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 __bool__ ; 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
Vincent Delecroix (2017-05-03): faster integer-rational comparison
Vincent Klein (2017-05-11): add __mpq__() to class Rational
Vincent Klein (2017-05-22): Rational constructor support gmpy2.mpq or gmpy2.mpz parameter. Add __mpz__ to class Rational.
- class sage.rings.rational.Q_to_Z[source]¶
Bases:
MapA morphism from \(\QQ\) to \(\ZZ\).
- section()[source]¶
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
>>> from sage.all import * >>> sage.rings.rational.Q_to_Z(QQ, ZZ).section() Natural morphism: From: Integer Ring To: Rational Field
sage.rings.rational.Q_to_Z(QQ, ZZ).section()
- class sage.rings.rational.Rational[source]¶
Bases:
FieldElementA rational number.
Rational numbers are implemented using the GMP C library.
EXAMPLES:
sage: a = -2/3 sage: type(a) <class '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)) # needs sage.rings.number_field 5/2 sage: Rational(AA(209735/343 - 17910/49*golden_ratio).nth_root(3) # needs sage.rings.number_field sage.symbolic ....: + 3*AA(golden_ratio)) 53/7 sage: QQ(float(1.5)) 3/2 sage: QQ(RDF(1.2)) 6/5
>>> from sage.all import * >>> a = -Integer(2)/Integer(3) >>> type(a) <class 'sage.rings.rational.Rational'> >>> parent(a) Rational Field >>> Rational('1/0') Traceback (most recent call last): ... TypeError: unable to convert '1/0' to a rational >>> Rational(RealNumber('1.5')) 3/2 >>> Rational('9/6') 3/2 >>> Rational((Integer(2)**Integer(99),Integer(2)**Integer(100))) 1/2 >>> Rational(("2", "10"), Integer(16)) 1/8 >>> Rational(QQbar(Integer(125)/Integer(8)).nth_root(Integer(3))) # needs sage.rings.number_field 5/2 >>> Rational(AA(Integer(209735)/Integer(343) - Integer(17910)/Integer(49)*golden_ratio).nth_root(Integer(3)) # needs sage.rings.number_field sage.symbolic ... + Integer(3)*AA(golden_ratio)) 53/7 >>> QQ(float(RealNumber('1.5'))) 3/2 >>> QQ(RDF(RealNumber('1.2'))) 6/5
a = -2/3 type(a) parent(a) Rational('1/0') Rational(1.5) Rational('9/6') Rational((2^99,2^100)) Rational(("2", "10"), 16) Rational(QQbar(125/8).nth_root(3)) # needs sage.rings.number_field Rational(AA(209735/343 - 17910/49*golden_ratio).nth_root(3) # needs sage.rings.number_field sage.symbolic + 3*AA(golden_ratio)) QQ(float(1.5)) QQ(RDF(1.2))Conversion from fractions:
sage: import fractions sage: f = fractions.Fraction(1r, 2r) sage: Rational(f) 1/2
>>> from sage.all import * >>> import fractions >>> f = fractions.Fraction(1, 2) >>> Rational(f) 1/2
import fractions f = fractions.Fraction(1r, 2r) Rational(f)
Conversion from PARI:
sage: Rational(pari('-939082/3992923')) # needs sage.libs.pari -939082/3992923 sage: Rational(pari('Pol([-1/2])')) #9595 # needs sage.libs.pari -1/2
>>> from sage.all import * >>> Rational(pari('-939082/3992923')) # needs sage.libs.pari -939082/3992923 >>> Rational(pari('Pol([-1/2])')) #9595 # needs sage.libs.pari -1/2
Rational(pari('-939082/3992923')) # needs sage.libs.pari Rational(pari('Pol([-1/2])')) #9595 # needs sage.libs.pariConversions from numpy:
sage: # needs numpy sage: import numpy as np sage: QQ(np.int8('-15')) -15 sage: QQ(np.int16('-32')) -32 sage: QQ(np.int32('-19')) -19 sage: QQ(np.uint32('1412')) 1412 sage: QQ(np.float16('12')) # needs numpy 12
>>> from sage.all import * >>> # needs numpy >>> import numpy as np >>> QQ(np.int8('-15')) -15 >>> QQ(np.int16('-32')) -32 >>> QQ(np.int32('-19')) -19 >>> QQ(np.uint32('1412')) 1412 >>> QQ(np.float16('12')) # needs numpy 12
# needs numpy import numpy as np QQ(np.int8('-15')) QQ(np.int16('-32')) QQ(np.int32('-19')) QQ(np.uint32('1412')) QQ(np.float16('12')) # needs numpyConversions from gmpy2:
sage: from gmpy2 import * sage: QQ(mpq('3/4')) 3/4 sage: QQ(mpz(42)) 42 sage: Rational(mpq(2/3)) 2/3 sage: Rational(mpz(5)) 5
>>> from sage.all import * >>> from gmpy2 import * >>> QQ(mpq('3/4')) 3/4 >>> QQ(mpz(Integer(42))) 42 >>> Rational(mpq(Integer(2)/Integer(3))) 2/3 >>> Rational(mpz(Integer(5))) 5
from gmpy2 import * QQ(mpq('3/4')) QQ(mpz(42)) Rational(mpq(2/3)) Rational(mpz(5))- absolute_norm()[source]¶
Return 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
>>> from sage.all import * >>> (Integer(6)/Integer(5)).absolute_norm() 6/5 >>> QQ(Integer(7)/Integer(5)).absolute_norm() 7/5
(6/5).absolute_norm() QQ(7/5).absolute_norm()
- additive_order()[source]¶
Return the additive order of
self.OUTPUT: integer or infinity
EXAMPLES:
sage: QQ(0).additive_order() 1 sage: QQ(1).additive_order() +Infinity
>>> from sage.all import * >>> QQ(Integer(0)).additive_order() 1 >>> QQ(Integer(1)).additive_order() +Infinity
QQ(0).additive_order() QQ(1).additive_order()
- as_integer_ratio()[source]¶
Return the pair
(self.numerator(), self.denominator()).EXAMPLES:
sage: x = -12/29 sage: x.as_integer_ratio() (-12, 29)
>>> from sage.all import * >>> x = -Integer(12)/Integer(29) >>> x.as_integer_ratio() (-12, 29)
x = -12/29 x.as_integer_ratio()
- ceil()[source]¶
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
>>> from sage.all import * >>> n = Integer(5)/Integer(3); n.ceil() 2 >>> n = -Integer(17)/Integer(19); n.ceil() 0 >>> n = -Integer(7)/Integer(2); n.ceil() -3 >>> n = Integer(7)/Integer(2); n.ceil() 4 >>> n = Integer(10)/Integer(2); n.ceil() 5
n = 5/3; n.ceil() n = -17/19; n.ceil() n = -7/2; n.ceil() n = 7/2; n.ceil() n = 10/2; n.ceil()
- charpoly(var='x')[source]¶
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– string
OUTPUT: polynomial
EXAMPLES:
sage: (1/3).charpoly('x') x - 1/3
>>> from sage.all import * >>> (Integer(1)/Integer(3)).charpoly('x') x - 1/3
(1/3).charpoly('x')The default is
var='x'. (Issue #20967):sage: a = QQ(2); a.charpoly('x') x - 2
>>> from sage.all import * >>> a = QQ(Integer(2)); a.charpoly('x') x - 2
a = QQ(2); a.charpoly('x')AUTHORS:
Craig Citro
- conjugate()[source]¶
Return the complex conjugate of this rational number, which is the number itself.
EXAMPLES:
sage: n = 23/11 sage: n.conjugate() 23/11
>>> from sage.all import * >>> n = Integer(23)/Integer(11) >>> n.conjugate() 23/11
n = 23/11 n.conjugate()
- content(other)[source]¶
Return the content of
selfandother, i.e., the unique positive rational number \(c\) such thatself/candother/care coprime integers.othercan 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
>>> from sage.all import * >>> a = Integer(2)/Integer(3) >>> a.content(Integer(2)/Integer(3)) 2/3 >>> a.content(Integer(1)/Integer(5)) 1/15 >>> a.content([Integer(2)/Integer(5), Integer(4)/Integer(9)]) 2/45
a = 2/3 a.content(2/3) a.content(1/5) a.content([2/5, 4/9])
- continued_fraction()[source]¶
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) # needs sage.rings.real_mpfr 3.141592920 sage: pi.n(digits=10) # needs sage.rings.real_mpfr sage.symbolic 3.141592654
>>> from sage.all import * >>> (Integer(641)/Integer(472)).continued_fraction() [1; 2, 1, 3, 1, 4, 1, 5] >>> a = (Integer(355)/Integer(113)).continued_fraction(); a [3; 7, 16] >>> a.n(digits=Integer(10)) # needs sage.rings.real_mpfr 3.141592920 >>> pi.n(digits=Integer(10)) # needs sage.rings.real_mpfr sage.symbolic 3.141592654
(641/472).continued_fraction() a = (355/113).continued_fraction(); a a.n(digits=10) # needs sage.rings.real_mpfr pi.n(digits=10) # needs sage.rings.real_mpfr sage.symbolic
It’s almost pi!
- continued_fraction_list(type='std')[source]¶
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() # needs sage.libs.pari [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]
>>> from sage.all import * >>> (Integer(13)/Integer(9)).continued_fraction_list() [1, 2, 4] >>> Integer(1) + Integer(1)/(Integer(2) + Integer(1)/Integer(4)) 13/9 >>> (Integer(225)/Integer(157)).continued_fraction_list() [1, 2, 3, 4, 5] >>> Integer(1) + Integer(1)/(Integer(2) + Integer(1)/(Integer(3) + Integer(1)/(Integer(4) + Integer(1)/Integer(5)))) 225/157 >>> (fibonacci(Integer(20))/fibonacci(Integer(19))).continued_fraction_list() # needs sage.libs.pari [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2] >>> (-Integer(1)/Integer(3)).continued_fraction_list() [-1, 1, 2]
(13/9).continued_fraction_list() 1 + 1/(2 + 1/4) (225/157).continued_fraction_list() 1 + 1/(2 + 1/(3 + 1/(4 + 1/5))) (fibonacci(20)/fibonacci(19)).continued_fraction_list() # needs sage.libs.pari (-1/3).continued_fraction_list()
Check that the partial quotients of an integer
nis simply[n]:sage: QQ(1).continued_fraction_list() [1] sage: QQ(0).continued_fraction_list() [0] sage: QQ(-1).continued_fraction_list() [-1]
>>> from sage.all import * >>> QQ(Integer(1)).continued_fraction_list() [1] >>> QQ(Integer(0)).continued_fraction_list() [0] >>> QQ(-Integer(1)).continued_fraction_list() [-1]
QQ(1).continued_fraction_list() QQ(0).continued_fraction_list() QQ(-1).continued_fraction_list()
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
>>> from sage.all import * >>> (Integer(11)/Integer(19)).continued_fraction_list("hj") [1, 3, 2, 3, 2] >>> Integer(1) - Integer(1)/(Integer(3) - Integer(1)/(Integer(2) - Integer(1)/(Integer(3) - Integer(1)/Integer(2)))) 11/19 >>> (Integer(225)/Integer(137)).continued_fraction_list("hj") [2, 3, 5, 10] >>> Integer(2) - Integer(1)/(Integer(3) - Integer(1)/(Integer(5) - Integer(1)/Integer(10))) 225/137 >>> (-Integer(23)/Integer(19)).continued_fraction_list("hj") [-1, 5, 4] >>> -Integer(1) - Integer(1)/(Integer(5) - Integer(1)/Integer(4)) -23/19
(11/19).continued_fraction_list("hj") 1 - 1/(3 - 1/(2 - 1/(3 - 1/2))) (225/137).continued_fraction_list("hj") 2 - 1/(3 - 1/(5 - 1/10)) (-23/19).continued_fraction_list("hj") -1 - 1/(5 - 1/4)
- denom()[source]¶
Return the denominator of this rational number.
denom()is an alias ofdenominator().EXAMPLES:
sage: x = -5/11 sage: x.denominator() 11 sage: x = 9/3 sage: x.denominator() 1 sage: x = 5/13 sage: x.denom() 13
>>> from sage.all import * >>> x = -Integer(5)/Integer(11) >>> x.denominator() 11 >>> x = Integer(9)/Integer(3) >>> x.denominator() 1 >>> x = Integer(5)/Integer(13) >>> x.denom() 13
x = -5/11 x.denominator() x = 9/3 x.denominator() x = 5/13 x.denom()
- denominator()[source]¶
Return the denominator of this rational number.
denom()is an alias ofdenominator().EXAMPLES:
sage: x = -5/11 sage: x.denominator() 11 sage: x = 9/3 sage: x.denominator() 1 sage: x = 5/13 sage: x.denom() 13
>>> from sage.all import * >>> x = -Integer(5)/Integer(11) >>> x.denominator() 11 >>> x = Integer(9)/Integer(3) >>> x.denominator() 1 >>> x = Integer(5)/Integer(13) >>> x.denom() 13
x = -5/11 x.denominator() x = 9/3 x.denominator() x = 5/13 x.denom()
- factor()[source]¶
Return the factorization of this rational number.
OUTPUT: factorization
EXAMPLES:
sage: (-4/17).factor() -1 * 2^2 * 17^-1
>>> from sage.all import * >>> (-Integer(4)/Integer(17)).factor() -1 * 2^2 * 17^-1
(-4/17).factor()
Trying to factor 0 gives an arithmetic error:
sage: (0/1).factor() Traceback (most recent call last): ... ArithmeticError: factorization of 0 is not defined
>>> from sage.all import * >>> (Integer(0)/Integer(1)).factor() Traceback (most recent call last): ... ArithmeticError: factorization of 0 is not defined
(0/1).factor()
- floor()[source]¶
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
>>> from sage.all import * >>> n = Integer(5)/Integer(3); n.floor() 1 >>> n = -Integer(17)/Integer(19); n.floor() -1 >>> n = -Integer(7)/Integer(2); n.floor() -4 >>> n = Integer(7)/Integer(2); n.floor() 3 >>> n = Integer(10)/Integer(2); n.floor() 5
n = 5/3; n.floor() n = -17/19; n.floor() n = -7/2; n.floor() n = 7/2; n.floor() n = 10/2; n.floor()
- gamma(prec=None)[source]¶
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 keywordprec.EXAMPLES:
sage: # needs sage.symbolic 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)
>>> from sage.all import * >>> # needs sage.symbolic >>> gamma(Integer(1)/Integer(2)) sqrt(pi) >>> gamma(Integer(7)/Integer(2)) 15/8*sqrt(pi) >>> gamma(-Integer(3)/Integer(2)) 4/3*sqrt(pi) >>> gamma(Integer(6)/Integer(1)) 120 >>> gamma(Integer(1)/Integer(3)) gamma(1/3)
# needs sage.symbolic gamma(1/2) gamma(7/2) gamma(-3/2) gamma(6/1) gamma(1/3)
This function accepts an optional precision argument:
sage: (1/3).gamma(prec=100) # needs sage.rings.real_mpfr 2.6789385347077476336556929410 sage: (1/2).gamma(prec=100) # needs sage.rings.real_mpfr 1.7724538509055160272981674833
>>> from sage.all import * >>> (Integer(1)/Integer(3)).gamma(prec=Integer(100)) # needs sage.rings.real_mpfr 2.6789385347077476336556929410 >>> (Integer(1)/Integer(2)).gamma(prec=Integer(100)) # needs sage.rings.real_mpfr 1.7724538509055160272981674833
(1/3).gamma(prec=100) # needs sage.rings.real_mpfr (1/2).gamma(prec=100) # needs sage.rings.real_mpfr
- global_height(prec=None)[source]¶
Return the absolute logarithmic height of this rational number.
INPUT:
prec– integer (default: defaultRealFieldprecision); desired floating point 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: # needs sage.rings.real_mpfr 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
>>> from sage.all import * >>> # needs sage.rings.real_mpfr >>> a = QQ(Integer(6)/Integer(25)) >>> a.global_height_arch() + a.global_height_non_arch() 3.21887582486820 >>> a.global_height() 3.21887582486820 >>> (Integer(1)/a).global_height() 3.21887582486820 >>> QQ(Integer(0)).global_height() 0.000000000000000 >>> QQ(Integer(1)).global_height() 0.000000000000000
# needs sage.rings.real_mpfr a = QQ(6/25) a.global_height_arch() + a.global_height_non_arch() a.global_height() (1/a).global_height() QQ(0).global_height() QQ(1).global_height()
- global_height_arch(prec=None)[source]¶
Return the total archimedean component of the height of this rational number.
INPUT:
prec– integer (default: defaultRealFieldprecision); desired floating point 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() # needs sage.rings.real_mpfr 0.000000000000000 sage: (1/a).global_height_arch() # needs sage.rings.real_mpfr 1.42711635564015 sage: (1/a).global_height_arch(100) # needs sage.rings.real_mpfr 1.4271163556401457483890413081
>>> from sage.all import * >>> a = QQ(Integer(6)/Integer(25)) >>> a.global_height_arch() # needs sage.rings.real_mpfr 0.000000000000000 >>> (Integer(1)/a).global_height_arch() # needs sage.rings.real_mpfr 1.42711635564015 >>> (Integer(1)/a).global_height_arch(Integer(100)) # needs sage.rings.real_mpfr 1.4271163556401457483890413081
a = QQ(6/25) a.global_height_arch() # needs sage.rings.real_mpfr (1/a).global_height_arch() # needs sage.rings.real_mpfr (1/a).global_height_arch(100) # needs sage.rings.real_mpfr
- global_height_non_arch(prec=None)[source]¶
Return the total non-archimedean component of the height of this rational number.
INPUT:
prec– integer (default: defaultRealFieldprecision); desired floating point 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() # needs sage.rings.real_mpfr 1.79175946922805 sage: [a.local_height(p) for p in a.support()] # needs sage.rings.real_mpfr [0.693147180559945, 1.09861228866811, 0.000000000000000] sage: sum([a.local_height(p) for p in a.support()]) # needs sage.rings.real_mpfr 1.79175946922805
>>> from sage.all import * >>> a = QQ(Integer(5)/Integer(6)) >>> a.support() [2, 3, 5] >>> a.global_height_non_arch() # needs sage.rings.real_mpfr 1.79175946922805 >>> [a.local_height(p) for p in a.support()] # needs sage.rings.real_mpfr [0.693147180559945, 1.09861228866811, 0.000000000000000] >>> sum([a.local_height(p) for p in a.support()]) # needs sage.rings.real_mpfr 1.79175946922805
a = QQ(5/6) a.support() a.global_height_non_arch() # needs sage.rings.real_mpfr [a.local_height(p) for p in a.support()] # needs sage.rings.real_mpfr sum([a.local_height(p) for p in a.support()]) # needs sage.rings.real_mpfr
- height()[source]¶
The max absolute value of the numerator and denominator of
self, as anInteger.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
>>> from sage.all import * >>> a = Integer(2)/Integer(3) >>> a.height() 3 >>> a = Integer(34)/Integer(3) >>> a.height() 34 >>> a = -Integer(97)/Integer(4) >>> a.height() 97
a = 2/3 a.height() a = 34/3 a.height() a = -97/4 a.height()
AUTHORS:
Naqi Jaffery (2006-03-05): examples
Note
For the logarithmic height, use
global_height().
- imag()[source]¶
Return the imaginary part of
self, which is zero.EXAMPLES:
sage: (1/239).imag() 0
>>> from sage.all import * >>> (Integer(1)/Integer(239)).imag() 0
(1/239).imag()
- is_S_integral(S=[])[source]¶
Determine if the rational number is
S-integral.xisS-integral ifx.valuation(p)>=0for allpnot inS, i.e., the denominator ofxis divisible only by the primes inS.INPUT:
S– list or tuple of primes
OUTPUT: boolean
Note
Primality of the entries in
Sis 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]
>>> from sage.all import * >>> QQ(Integer(1)/Integer(2)).is_S_integral() False >>> QQ(Integer(1)/Integer(2)).is_S_integral([Integer(2)]) True >>> [a for a in range(Integer(1),Integer(11)) if QQ(Integer(101)/a).is_S_integral([Integer(2),Integer(5)])] [1, 2, 4, 5, 8, 10]
QQ(1/2).is_S_integral() QQ(1/2).is_S_integral([2]) [a for a in range(1,11) if QQ(101/a).is_S_integral([2,5])]
- is_S_unit(S=None)[source]¶
Determine if the rational number is an
S-unit.xis anS-unit ifx.valuation(p)==0for allpnot inS, i.e., the numerator and denominator ofxare divisible only by the primes in \(S\).INPUT:
S– list or tuple of primes
OUTPUT: boolean
Note
Primality of the entries in
Sis 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]
>>> from sage.all import * >>> QQ(Integer(1)/Integer(2)).is_S_unit() False >>> QQ(Integer(1)/Integer(2)).is_S_unit([Integer(2)]) True >>> [a for a in range(Integer(1),Integer(11)) if QQ(Integer(10)/a).is_S_unit([Integer(2),Integer(5)])] [1, 2, 4, 5, 8, 10]
QQ(1/2).is_S_unit() QQ(1/2).is_S_unit([2]) [a for a in range(1,11) if QQ(10/a).is_S_unit([2,5])]
- is_integer()[source]¶
Determine if a rational number is integral (i.e., is in \(\ZZ\)).
OUTPUT: boolean
EXAMPLES:
sage: QQ(1/2).is_integral() False sage: QQ(4/4).is_integral() True
>>> from sage.all import * >>> QQ(Integer(1)/Integer(2)).is_integral() False >>> QQ(Integer(4)/Integer(4)).is_integral() True
QQ(1/2).is_integral() QQ(4/4).is_integral()
- is_integral()[source]¶
Determine if a rational number is integral (i.e., is in \(\ZZ\)).
OUTPUT: boolean
EXAMPLES:
sage: QQ(1/2).is_integral() False sage: QQ(4/4).is_integral() True
>>> from sage.all import * >>> QQ(Integer(1)/Integer(2)).is_integral() False >>> QQ(Integer(4)/Integer(4)).is_integral() True
QQ(1/2).is_integral() QQ(4/4).is_integral()
- is_norm(L, element=False, proof=True)[source]¶
Determine whether
selfis the norm of an element ofL.INPUT:
L– a number fieldelement– boolean (default:False); whether to also output an element of whichselfis a normproof– ifTrue, then the output is correct unconditionally; ifFalse, then the output assumes GRH
OUTPUT:
If element is
False, then the output is a booleanB, which isTrueif and only ifselfis the norm of an element ofL. IfelementisFalse, then the output is a pair(B, x), whereBis as above. IfBisTrue, thenxan element ofLsuch thatself == x.norm(). Otherwise,x is None.ALGORITHM:
Uses the PARI function pari:bnfisnorm. See
_bnfisnorm().EXAMPLES:
sage: # needs sage.rings.number_field sage: x = polygen(QQ, 'x') 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)
>>> from sage.all import * >>> # needs sage.rings.number_field >>> x = polygen(QQ, 'x') >>> K = NumberField(x**Integer(2) - Integer(2), 'beta') >>> (Integer(1)/Integer(7)).is_norm(K) True >>> (Integer(1)/Integer(10)).is_norm(K) False >>> Integer(0).is_norm(K) True >>> (Integer(1)/Integer(7)).is_norm(K, element=True) (True, 1/7*beta + 3/7) >>> (Integer(1)/Integer(10)).is_norm(K, element=True) (False, None) >>> (Integer(1)/Integer(691)).is_norm(QQ, element=True) (True, 1/691)
# needs sage.rings.number_field x = polygen(QQ, 'x') K = NumberField(x^2 - 2, 'beta') (1/7).is_norm(K) (1/10).is_norm(K) 0.is_norm(K) (1/7).is_norm(K, element=True) (1/10).is_norm(K, element=True) (1/691).is_norm(QQ, element=True)
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) # needs sage.rings.number_field sage: B # needs sage.rings.number_field True sage: e.norm() # needs sage.rings.number_field 1/5
>>> from sage.all import * >>> B, e = (Integer(1)/Integer(5)).is_norm(QuadraticField(Integer(5)/Integer(4), 'a'), element=True) # needs sage.rings.number_field >>> B # needs sage.rings.number_field True >>> e.norm() # needs sage.rings.number_field 1/5
B, e = (1/5).is_norm(QuadraticField(5/4, 'a'), element=True) # needs sage.rings.number_field B # needs sage.rings.number_field e.norm() # needs sage.rings.number_field
A non-Galois number field:
sage: # needs sage.rings.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) # needs sage.groups 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
>>> from sage.all import * >>> # needs sage.rings.number_field >>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1) >>> B, e = (Integer(3)/Integer(5)).is_norm(K, element=True); B True >>> e.norm() 3/5 >>> Integer(7).is_norm(K) # needs sage.groups Traceback (most recent call last): ... NotImplementedError: is_norm is not implemented unconditionally for norms from non-Galois number fields >>> Integer(7).is_norm(K, proof=False) False
# needs sage.rings.number_field K.<a> = NumberField(x^3 - 2) B, e = (3/5).is_norm(K, element=True); B e.norm() 7.is_norm(K) # needs sage.groups 7.is_norm(K, proof=False)
AUTHORS:
Craig Citro (2008-04-05)
Marco Streng (2010-12-03)
- is_nth_power(n)[source]¶
Return
Trueifselfis an \(n\)-th power, elseFalse.INPUT:
n– integer (must fit in Cinttype)
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
>>> from sage.all import * >>> QQ(Integer(25)/Integer(4)).is_nth_power(Integer(2)) True >>> QQ(Integer(125)/Integer(8)).is_nth_power(Integer(3)) True >>> QQ(-Integer(125)/Integer(8)).is_nth_power(Integer(3)) True >>> QQ(Integer(25)/Integer(4)).is_nth_power(-Integer(2)) True >>> QQ(Integer(9)/Integer(2)).is_nth_power(Integer(2)) False >>> QQ(-Integer(25)).is_nth_power(Integer(2)) False
QQ(25/4).is_nth_power(2) QQ(125/8).is_nth_power(3) QQ(-125/8).is_nth_power(3) QQ(25/4).is_nth_power(-2) QQ(9/2).is_nth_power(2) QQ(-25).is_nth_power(2)
- is_one()[source]¶
Determine if a rational number is one.
OUTPUT: boolean
EXAMPLES:
sage: QQ(1/2).is_one() False sage: QQ(4/4).is_one() True
>>> from sage.all import * >>> QQ(Integer(1)/Integer(2)).is_one() False >>> QQ(Integer(4)/Integer(4)).is_one() True
QQ(1/2).is_one() QQ(4/4).is_one()
- is_padic_square(p, check=True)[source]¶
Determines whether this rational number is a square in \(\QQ_p\) (or in \(R\) when
p = infinity).INPUT:
p– a prime number, orinfinitycheck– boolean (default:True); check if \(p\) is prime
EXAMPLES:
sage: QQ(2).is_padic_square(7) True sage: QQ(98).is_padic_square(7) True sage: QQ(2).is_padic_square(5) False
>>> from sage.all import * >>> QQ(Integer(2)).is_padic_square(Integer(7)) True >>> QQ(Integer(98)).is_padic_square(Integer(7)) True >>> QQ(Integer(2)).is_padic_square(Integer(5)) False
QQ(2).is_padic_square(7) QQ(98).is_padic_square(7) QQ(2).is_padic_square(5)
- is_perfect_power(expected_value=False)[source]¶
Return
Trueifselfis a perfect power.INPUT:
expected_value– boolean; whether or not this rational is expected to be a perfect power. This does not affect the correctness of the output, only the runtime.
If
expected_valueisFalse(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
>>> from sage.all import * >>> (Integer(4)/Integer(9)).is_perfect_power() True >>> (Integer(144)/Integer(1)).is_perfect_power() True >>> (Integer(4)/Integer(3)).is_perfect_power() False >>> (Integer(2)/Integer(27)).is_perfect_power() False >>> (Integer(4)/Integer(27)).is_perfect_power() False >>> (-Integer(1)/Integer(25)).is_perfect_power() False >>> (-Integer(1)/Integer(27)).is_perfect_power() True >>> (Integer(0)/Integer(1)).is_perfect_power() True
(4/9).is_perfect_power() (144/1).is_perfect_power() (4/3).is_perfect_power() (2/27).is_perfect_power() (4/27).is_perfect_power() (-1/25).is_perfect_power() (-1/27).is_perfect_power() (0/1).is_perfect_power()
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
>>> from sage.all import * >>> (-Integer(1)/Integer(27)).is_perfect_power(True) True >>> (-Integer(1)/Integer(25)).is_perfect_power(True) False >>> (Integer(2)/Integer(27)).is_perfect_power(True) False >>> (Integer(144)/Integer(1)).is_perfect_power(True) True
(-1/27).is_perfect_power(True) (-1/25).is_perfect_power(True) (2/27).is_perfect_power(True) (144/1).is_perfect_power(True)
This test makes sure we workaround a bug in GMP (see Issue #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)] []
>>> from sage.all import * >>> [-a for a in srange(Integer(100)) if not QQ(-a**Integer(3)).is_perfect_power()] [] >>> [-a for a in srange(Integer(100)) if not QQ(-a**Integer(3)).is_perfect_power(True)] []
[-a for a in srange(100) if not QQ(-a^3).is_perfect_power()] [-a for a in srange(100) if not QQ(-a^3).is_perfect_power(True)]
- is_rational()[source]¶
Return
Truesince this is a rational number.EXAMPLES:
sage: (3/4).is_rational() True
>>> from sage.all import * >>> (Integer(3)/Integer(4)).is_rational() True
(3/4).is_rational()
- is_square()[source]¶
Return whether or not this rational number is a square.
OUTPUT: boolean
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
>>> from sage.all import * >>> x = Integer(9)/Integer(4) >>> x.is_square() True >>> x = (Integer(7)/Integer(53))**Integer(100) >>> x.is_square() True >>> x = Integer(4)/Integer(3) >>> x.is_square() False >>> x = -Integer(1)/Integer(4) >>> x.is_square() False
x = 9/4 x.is_square() x = (7/53)^100 x.is_square() x = 4/3 x.is_square() x = -1/4 x.is_square()
- list()[source]¶
Return a list with the rational element in it, to be compatible with the method for number fields.
OUTPUT: the list
[self]EXAMPLES:
sage: m = 5/3 sage: m.list() [5/3]
>>> from sage.all import * >>> m = Integer(5)/Integer(3) >>> m.list() [5/3]
m = 5/3 m.list()
- local_height(p, prec=None)[source]¶
Return the local height of this rational number at the prime \(p\).
INPUT:
p– a prime numberprec– integer (default: defaultRealFieldprecision); desired floating point 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) # needs sage.rings.real_mpfr 0.693147180559945 sage: a.local_height(3) # needs sage.rings.real_mpfr 1.09861228866811 sage: a.local_height(5) # needs sage.rings.real_mpfr 0.000000000000000
>>> from sage.all import * >>> a = QQ(Integer(25)/Integer(6)) >>> a.local_height(Integer(2)) # needs sage.rings.real_mpfr 0.693147180559945 >>> a.local_height(Integer(3)) # needs sage.rings.real_mpfr 1.09861228866811 >>> a.local_height(Integer(5)) # needs sage.rings.real_mpfr 0.000000000000000
a = QQ(25/6) a.local_height(2) # needs sage.rings.real_mpfr a.local_height(3) # needs sage.rings.real_mpfr a.local_height(5) # needs sage.rings.real_mpfr
- local_height_arch(prec=None)[source]¶
Return the Archimedean local height of this rational number at the infinite place.
INPUT:
prec– integer (default: defaultRealFieldprecision); desired floating point 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() # needs sage.rings.real_mpfr 0.000000000000000 sage: (1/a).local_height_arch() # needs sage.rings.real_mpfr 1.42711635564015 sage: (1/a).local_height_arch(100) # needs sage.rings.real_mpfr 1.4271163556401457483890413081
>>> from sage.all import * >>> a = QQ(Integer(6)/Integer(25)) >>> a.local_height_arch() # needs sage.rings.real_mpfr 0.000000000000000 >>> (Integer(1)/a).local_height_arch() # needs sage.rings.real_mpfr 1.42711635564015 >>> (Integer(1)/a).local_height_arch(Integer(100)) # needs sage.rings.real_mpfr 1.4271163556401457483890413081
a = QQ(6/25) a.local_height_arch() # needs sage.rings.real_mpfr (1/a).local_height_arch() # needs sage.rings.real_mpfr (1/a).local_height_arch(100) # needs sage.rings.real_mpfr
- log(m=None, prec=None)[source]¶
Return the log of
self.INPUT:
m– the base (default: natural log base e)prec– integer (optional); the precision in bits
OUTPUT:
When
precis not given, the log as an element in symbolic ring unless the logarithm is exact. Otherwise the log is aRealFieldapproximation toprecbit precision.EXAMPLES:
sage: (124/345).log(5) # needs sage.symbolic log(124/345)/log(5) sage: (124/345).log(5, 100) # needs sage.rings.real_mpfr -0.63578895682825611710391773754 sage: log(QQ(125)) # needs sage.symbolic 3*log(5) sage: log(QQ(125), 5) 3 sage: log(QQ(125), 3) # needs sage.symbolic 3*log(5)/log(3) sage: QQ(8).log(1/2) -3 sage: (1/8).log(1/2) 3 sage: (1/2).log(1/8) 1/3 sage: (1/2).log(8) -1/3 sage: (16/81).log(8/27) # needs sage.libs.pari 4/3 sage: (8/27).log(16/81) # needs sage.libs.pari 3/4 sage: log(27/8, 16/81) # needs sage.libs.pari -3/4 sage: log(16/81, 27/8) # needs sage.libs.pari -4/3 sage: (125/8).log(5/2) # needs sage.libs.pari 3 sage: (125/8).log(5/2, prec=53) # needs sage.rings.real_mpfr 3.00000000000000
>>> from sage.all import * >>> (Integer(124)/Integer(345)).log(Integer(5)) # needs sage.symbolic log(124/345)/log(5) >>> (Integer(124)/Integer(345)).log(Integer(5), Integer(100)) # needs sage.rings.real_mpfr -0.63578895682825611710391773754 >>> log(QQ(Integer(125))) # needs sage.symbolic 3*log(5) >>> log(QQ(Integer(125)), Integer(5)) 3 >>> log(QQ(Integer(125)), Integer(3)) # needs sage.symbolic 3*log(5)/log(3) >>> QQ(Integer(8)).log(Integer(1)/Integer(2)) -3 >>> (Integer(1)/Integer(8)).log(Integer(1)/Integer(2)) 3 >>> (Integer(1)/Integer(2)).log(Integer(1)/Integer(8)) 1/3 >>> (Integer(1)/Integer(2)).log(Integer(8)) -1/3 >>> (Integer(16)/Integer(81)).log(Integer(8)/Integer(27)) # needs sage.libs.pari 4/3 >>> (Integer(8)/Integer(27)).log(Integer(16)/Integer(81)) # needs sage.libs.pari 3/4 >>> log(Integer(27)/Integer(8), Integer(16)/Integer(81)) # needs sage.libs.pari -3/4 >>> log(Integer(16)/Integer(81), Integer(27)/Integer(8)) # needs sage.libs.pari -4/3 >>> (Integer(125)/Integer(8)).log(Integer(5)/Integer(2)) # needs sage.libs.pari 3 >>> (Integer(125)/Integer(8)).log(Integer(5)/Integer(2), prec=Integer(53)) # needs sage.rings.real_mpfr 3.00000000000000
(124/345).log(5) # needs sage.symbolic (124/345).log(5, 100) # needs sage.rings.real_mpfr log(QQ(125)) # needs sage.symbolic log(QQ(125), 5) log(QQ(125), 3) # needs sage.symbolic QQ(8).log(1/2) (1/8).log(1/2) (1/2).log(1/8) (1/2).log(8) (16/81).log(8/27) # needs sage.libs.pari (8/27).log(16/81) # needs sage.libs.pari log(27/8, 16/81) # needs sage.libs.pari log(16/81, 27/8) # needs sage.libs.pari (125/8).log(5/2) # needs sage.libs.pari (125/8).log(5/2, prec=53) # needs sage.rings.real_mpfr
- minpoly(var='x')[source]¶
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– string
OUTPUT: polynomial
EXAMPLES:
sage: (1/3).minpoly() x - 1/3 sage: (1/3).minpoly('y') y - 1/3
>>> from sage.all import * >>> (Integer(1)/Integer(3)).minpoly() x - 1/3 >>> (Integer(1)/Integer(3)).minpoly('y') y - 1/3
(1/3).minpoly() (1/3).minpoly('y')AUTHORS:
Craig Citro
- mod_ui(n)[source]¶
Return the remainder upon division of
selfby the unsigned long integern.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.
>>> from sage.all import * >>> (-Integer(4)/Integer(17)).mod_ui(Integer(3)) 1 >>> (-Integer(4)/Integer(17)).mod_ui(Integer(17)) Traceback (most recent call last): ... ArithmeticError: The inverse of 0 modulo 17 is not defined.
(-4/17).mod_ui(3) (-4/17).mod_ui(17)
- multiplicative_order()[source]¶
Return the multiplicative order of
self.OUTPUT: integer or
infinityEXAMPLES:
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
>>> from sage.all import * >>> QQ(Integer(1)).multiplicative_order() 1 >>> QQ('1/-1').multiplicative_order() 2 >>> QQ(Integer(0)).multiplicative_order() +Infinity >>> QQ('2/3').multiplicative_order() +Infinity >>> QQ('1/2').multiplicative_order() +Infinity
QQ(1).multiplicative_order() QQ('1/-1').multiplicative_order() QQ(0).multiplicative_order() QQ('2/3').multiplicative_order() QQ('1/2').multiplicative_order()
- norm()[source]¶
Return the norm from \(\QQ\) to \(\QQ\) of \(x\) (which is just \(x\)). This was added for compatibility with
NumberField.OUTPUT:
Rational– reference toselfEXAMPLES:
sage: (1/3).norm() 1/3
>>> from sage.all import * >>> (Integer(1)/Integer(3)).norm() 1/3
(1/3).norm()
AUTHORS:
Craig Citro
- nth_root(n)[source]¶
Compute the \(n\)-th root of
self, or raises aValueErrorifselfis not a perfect \(n\)-th power.INPUT:
n– integer (must fit in Cinttype)
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
>>> from sage.all import * >>> (Integer(25)/Integer(4)).nth_root(Integer(2)) 5/2 >>> (Integer(125)/Integer(8)).nth_root(Integer(3)) 5/2 >>> (-Integer(125)/Integer(8)).nth_root(Integer(3)) -5/2 >>> (Integer(25)/Integer(4)).nth_root(-Integer(2)) 2/5
(25/4).nth_root(2) (125/8).nth_root(3) (-125/8).nth_root(3) (25/4).nth_root(-2)
sage: (9/2).nth_root(2) Traceback (most recent call last): ... ValueError: not a perfect 2nd power
>>> from sage.all import * >>> (Integer(9)/Integer(2)).nth_root(Integer(2)) Traceback (most recent call last): ... ValueError: not a perfect 2nd power
(9/2).nth_root(2)
>>> from sage.all import * >>> (Integer(9)/Integer(2)).nth_root(Integer(2)) Traceback (most recent call last): ... ValueError: not a perfect 2nd power
(9/2).nth_root(2)
sage: (-25/4).nth_root(2) Traceback (most recent call last): ... ValueError: cannot take even root of negative number
>>> from sage.all import * >>> (-Integer(25)/Integer(4)).nth_root(Integer(2)) Traceback (most recent call last): ... ValueError: cannot take even root of negative number
(-25/4).nth_root(2)
>>> from sage.all import * >>> (-Integer(25)/Integer(4)).nth_root(Integer(2)) Traceback (most recent call last): ... ValueError: cannot take even root of negative number
(-25/4).nth_root(2)
- numer()[source]¶
Return the numerator of this rational number.
numer()is an alias ofnumerator().EXAMPLES:
sage: x = 5/11 sage: x.numerator() 5 sage: x = 9/3 sage: x.numerator() 3 sage: x = -5/11 sage: x.numer() -5
>>> from sage.all import * >>> x = Integer(5)/Integer(11) >>> x.numerator() 5 >>> x = Integer(9)/Integer(3) >>> x.numerator() 3 >>> x = -Integer(5)/Integer(11) >>> x.numer() -5
x = 5/11 x.numerator() x = 9/3 x.numerator() x = -5/11 x.numer()
- numerator()[source]¶
Return the numerator of this rational number.
numer()is an alias ofnumerator().EXAMPLES:
sage: x = 5/11 sage: x.numerator() 5 sage: x = 9/3 sage: x.numerator() 3 sage: x = -5/11 sage: x.numer() -5
>>> from sage.all import * >>> x = Integer(5)/Integer(11) >>> x.numerator() 5 >>> x = Integer(9)/Integer(3) >>> x.numerator() 3 >>> x = -Integer(5)/Integer(11) >>> x.numer() -5
x = 5/11 x.numerator() x = 9/3 x.numerator() x = -5/11 x.numer()
- ord(p)[source]¶
Return the power of
pin the factorization ofself.INPUT:
p– a prime number
OUTPUT:
(integer or infinity)
Infinityifselfis zero, otherwise the (positive or negative) integer \(e\) such thatself= \(m*p^e\) with \(m\) coprime to \(p\).Note
See also
val_unit()which returns the pair \((e,m)\). The functionord()is an alias forvaluation().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
>>> from sage.all import * >>> x = -Integer(5)/Integer(9) >>> x.valuation(Integer(5)) 1 >>> x.ord(Integer(5)) 1 >>> x.valuation(Integer(3)) -2 >>> x.valuation(Integer(2)) 0
x = -5/9 x.valuation(5) x.ord(5) x.valuation(3) x.valuation(2)
Some edge cases:
sage: (0/1).valuation(4) +Infinity sage: (7/16).valuation(4) -2
>>> from sage.all import * >>> (Integer(0)/Integer(1)).valuation(Integer(4)) +Infinity >>> (Integer(7)/Integer(16)).valuation(Integer(4)) -2
(0/1).valuation(4) (7/16).valuation(4)
- period()[source]¶
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 \(d=2^a 5^b m\) where \(m\) is coprime to 10, then \(s=\max(a,b)\) and \(t\) is the order of 10 modulo \(m\).
EXAMPLES:
sage: (1/7).period() # needs sage.libs.pari 6 sage: RR(1/7) # needs sage.rings.real_mpfr 0.142857142857143 sage: (1/8).period() # needs sage.libs.pari 1 sage: RR(1/8) # needs sage.rings.real_mpfr 0.125000000000000 sage: RR(1/6) # needs sage.rings.real_mpfr 0.166666666666667 sage: (1/6).period() # needs sage.libs.pari 1 sage: x = 333/106 sage: x.period() # needs sage.libs.pari 13 sage: RealField(200)(x) # needs sage.rings.real_mpfr 3.1415094339622641509433962264150943396226415094339622641509
>>> from sage.all import * >>> (Integer(1)/Integer(7)).period() # needs sage.libs.pari 6 >>> RR(Integer(1)/Integer(7)) # needs sage.rings.real_mpfr 0.142857142857143 >>> (Integer(1)/Integer(8)).period() # needs sage.libs.pari 1 >>> RR(Integer(1)/Integer(8)) # needs sage.rings.real_mpfr 0.125000000000000 >>> RR(Integer(1)/Integer(6)) # needs sage.rings.real_mpfr 0.166666666666667 >>> (Integer(1)/Integer(6)).period() # needs sage.libs.pari 1 >>> x = Integer(333)/Integer(106) >>> x.period() # needs sage.libs.pari 13 >>> RealField(Integer(200))(x) # needs sage.rings.real_mpfr 3.1415094339622641509433962264150943396226415094339622641509
(1/7).period() # needs sage.libs.pari RR(1/7) # needs sage.rings.real_mpfr (1/8).period() # needs sage.libs.pari RR(1/8) # needs sage.rings.real_mpfr RR(1/6) # needs sage.rings.real_mpfr (1/6).period() # needs sage.libs.pari x = 333/106 x.period() # needs sage.libs.pari RealField(200)(x) # needs sage.rings.real_mpfr
- prime_to_S_part(S=[])[source]¶
Return
selfwith all powers of all primes inSremoved.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
>>> from sage.all import * >>> QQ(Integer(3)/Integer(4)).prime_to_S_part() 3/4 >>> QQ(Integer(3)/Integer(4)).prime_to_S_part([Integer(2)]) 3 >>> QQ(-Integer(3)/Integer(4)).prime_to_S_part([Integer(3)]) -1/4 >>> QQ(Integer(700)/Integer(99)).prime_to_S_part([Integer(2),Integer(3),Integer(5)]) 7/11 >>> QQ(-Integer(700)/Integer(99)).prime_to_S_part([Integer(2),Integer(3),Integer(5)]) -7/11 >>> QQ(Integer(0)).prime_to_S_part([Integer(2),Integer(3),Integer(5)]) 0 >>> QQ(-Integer(700)/Integer(99)).prime_to_S_part([]) -700/99
QQ(3/4).prime_to_S_part() QQ(3/4).prime_to_S_part([2]) QQ(-3/4).prime_to_S_part([3]) QQ(700/99).prime_to_S_part([2,3,5]) QQ(-700/99).prime_to_S_part([2,3,5]) QQ(0).prime_to_S_part([2,3,5]) QQ(-700/99).prime_to_S_part([])
- real()[source]¶
Return the real part of
self, which isself.EXAMPLES:
sage: (1/2).real() 1/2
>>> from sage.all import * >>> (Integer(1)/Integer(2)).real() 1/2
(1/2).real()
- relative_norm()[source]¶
Return 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
>>> from sage.all import * >>> (Integer(6)/Integer(5)).relative_norm() 6/5 >>> QQ(Integer(7)/Integer(5)).relative_norm() 7/5
(6/5).relative_norm() QQ(7/5).relative_norm()
- round(mode=None)[source]¶
Return the nearest integer to
self, rounding away by default. Deprecation: in the future the default will be changed to rounding to even, for consistency with the builtin Pythonround().INPUT:
self– a rational numbermode– 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() doctest:...: DeprecationWarning: the default rounding for rationals, currently `away`, will be changed to `even`. See https://github.com/sagemath/sage/issues/35473 for details. 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
>>> from sage.all import * >>> (Integer(9)/Integer(2)).round() doctest:...: DeprecationWarning: the default rounding for rationals, currently `away`, will be changed to `even`. See https://github.com/sagemath/sage/issues/35473 for details. 5 >>> n = Integer(4)/Integer(3); n.round() 1 >>> n = -Integer(17)/Integer(4); n.round() -4 >>> n = -Integer(5)/Integer(2); n.round() -3 >>> n.round("away") -3 >>> n.round("up") -2 >>> n.round("down") -3 >>> n.round("even") -2 >>> n.round("odd") -3
(9/2).round() n = 4/3; n.round() n = -17/4; n.round() n = -5/2; n.round() n.round("away") n.round("up") n.round("down") n.round("even") n.round("odd")
- sign()[source]¶
Return 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
>>> from sage.all import * >>> (Integer(2)/Integer(3)).sign() 1 >>> (Integer(0)/Integer(3)).sign() 0 >>> (-Integer(1)/Integer(6)).sign() -1
(2/3).sign() (0/3).sign() (-1/6).sign()
- sqrt(prec=None, extend=True, all=False)[source]¶
The square root function.
INPUT:
prec– integer (default:None); ifNone, returns an exact square root; otherwise returns a numerical square root if necessary, to the given bits of precision.extend– boolean (default:True); ifTrue, return a square root in an extension ring, if necessary. Otherwise, raise aValueErrorif the square is not in the base ring. Ignored ifprecis notNone.all– boolean (default:False); ifTrue, return all square roots ofself(a list of length 0, 1, or 2)
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() # needs sage.symbolic 9*sqrt(1/5) sage: x = -81/3 sage: x.sqrt() # needs sage.symbolic 3*sqrt(-3)
>>> from sage.all import * >>> x = Integer(25)/Integer(9) >>> x.sqrt() 5/3 >>> sqrt(x) 5/3 >>> x = Integer(64)/Integer(4) >>> x.sqrt() 4 >>> x = Integer(100)/Integer(1) >>> x.sqrt() 10 >>> x.sqrt(all=True) [10, -10] >>> x = Integer(81)/Integer(5) >>> x.sqrt() # needs sage.symbolic 9*sqrt(1/5) >>> x = -Integer(81)/Integer(3) >>> x.sqrt() # needs sage.symbolic 3*sqrt(-3)
x = 25/9 x.sqrt() sqrt(x) x = 64/4 x.sqrt() x = 100/1 x.sqrt() x.sqrt(all=True) x = 81/5 x.sqrt() # needs sage.symbolic x = -81/3 x.sqrt() # needs sage.symbolic
sage: n = 2/3 sage: n.sqrt() # needs sage.symbolic sqrt(2/3) sage: # needs sage.rings.real_mpfr 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: sqrt(-2/3, prec=53) 0.816496580927726*I sage: sqrt(-2/3, prec=53, all=True) [0.816496580927726*I, -0.816496580927726*I] sage: n.sqrt(extend=False) Traceback (most recent call last): ... ValueError: square root of 2/3 not a rational number sage: n.sqrt(extend=False, all=True) [] sage: sqrt(-2/3, all=True) # needs sage.symbolic [sqrt(-2/3), -sqrt(-2/3)]
>>> from sage.all import * >>> n = Integer(2)/Integer(3) >>> n.sqrt() # needs sage.symbolic sqrt(2/3) >>> # needs sage.rings.real_mpfr >>> n.sqrt(prec=Integer(10)) 0.82 >>> n.sqrt(prec=Integer(100)) 0.81649658092772603273242802490 >>> n.sqrt(prec=Integer(100))**Integer(2) 0.66666666666666666666666666667 >>> n.sqrt(prec=Integer(53), all=True) [0.816496580927726, -0.816496580927726] >>> sqrt(-Integer(2)/Integer(3), prec=Integer(53)) 0.816496580927726*I >>> sqrt(-Integer(2)/Integer(3), prec=Integer(53), all=True) [0.816496580927726*I, -0.816496580927726*I] >>> n.sqrt(extend=False) Traceback (most recent call last): ... ValueError: square root of 2/3 not a rational number >>> n.sqrt(extend=False, all=True) [] >>> sqrt(-Integer(2)/Integer(3), all=True) # needs sage.symbolic [sqrt(-2/3), -sqrt(-2/3)]
n = 2/3 n.sqrt() # needs sage.symbolic # needs sage.rings.real_mpfr n.sqrt(prec=10) n.sqrt(prec=100) n.sqrt(prec=100)^2 n.sqrt(prec=53, all=True) sqrt(-2/3, prec=53) sqrt(-2/3, prec=53, all=True) n.sqrt(extend=False) n.sqrt(extend=False, all=True) sqrt(-2/3, all=True) # needs sage.symbolic
>>> from sage.all import * >>> n = Integer(2)/Integer(3) >>> n.sqrt() # needs sage.symbolic sqrt(2/3) >>> # needs sage.rings.real_mpfr >>> n.sqrt(prec=Integer(10)) 0.82 >>> n.sqrt(prec=Integer(100)) 0.81649658092772603273242802490 >>> n.sqrt(prec=Integer(100))**Integer(2) 0.66666666666666666666666666667 >>> n.sqrt(prec=Integer(53), all=True) [0.816496580927726, -0.816496580927726] >>> sqrt(-Integer(2)/Integer(3), prec=Integer(53)) 0.816496580927726*I >>> sqrt(-Integer(2)/Integer(3), prec=Integer(53), all=True) [0.816496580927726*I, -0.816496580927726*I] >>> n.sqrt(extend=False) Traceback (most recent call last): ... ValueError: square root of 2/3 not a rational number >>> n.sqrt(extend=False, all=True) [] >>> sqrt(-Integer(2)/Integer(3), all=True) # needs sage.symbolic [sqrt(-2/3), -sqrt(-2/3)]
n = 2/3 n.sqrt() # needs sage.symbolic # needs sage.rings.real_mpfr n.sqrt(prec=10) n.sqrt(prec=100) n.sqrt(prec=100)^2 n.sqrt(prec=53, all=True) sqrt(-2/3, prec=53) sqrt(-2/3, prec=53, all=True) n.sqrt(extend=False) n.sqrt(extend=False, all=True) sqrt(-2/3, all=True) # needs sage.symbolic
AUTHORS:
Naqi Jaffery (2006-03-05): some examples
- squarefree_part()[source]¶
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
>>> from sage.all import * >>> a = Integer(1)/Integer(2) >>> a.squarefree_part() 2 >>> b = a/a.squarefree_part() >>> b, b.is_square() (1/4, True) >>> a = Integer(24)/Integer(5) >>> a.squarefree_part() 30
a = 1/2 a.squarefree_part() b = a/a.squarefree_part() b, b.is_square() a = 24/5 a.squarefree_part()
- str(base=10)[source]¶
Return a string representation of
selfin the givenbase.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'
>>> from sage.all import * >>> (-Integer(4)/Integer(17)).str() '-4/17' >>> (-Integer(4)/Integer(17)).str(Integer(2)) '-100/10001'
(-4/17).str() (-4/17).str(2)
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
>>> from sage.all import * >>> (-Integer(4)/Integer(17)).str(Integer(40)) Traceback (most recent call last): ... ValueError: base (=40) must be between 2 and 36 >>> (-Integer(4)/Integer(17)).str(Integer(1)) Traceback (most recent call last): ... ValueError: base (=1) must be between 2 and 36
(-4/17).str(40) (-4/17).str(1)
- support()[source]¶
Return a sorted list of the primes where this rational number has nonzero 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]
>>> from sage.all import * >>> (-Integer(4)/Integer(17)).support() [2, 17]
(-4/17).support()
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.
>>> from sage.all import * >>> (Integer(0)/Integer(1)).support() Traceback (most recent call last): ... ArithmeticError: Support of 0 not defined.
(0/1).support()
- trace()[source]¶
Return the trace from \(\QQ\) to \(\QQ\) of \(x\) (which is just \(x\)). This was added for compatibility with
NumberFields.OUTPUT:
Rational– reference toselfEXAMPLES:
sage: (1/3).trace() 1/3
>>> from sage.all import * >>> (Integer(1)/Integer(3)).trace() 1/3
(1/3).trace()
AUTHORS:
Craig Citro
- trunc()[source]¶
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
>>> from sage.all import * >>> (Integer(5)/Integer(3)).trunc() 1 >>> (-Integer(5)/Integer(3)).trunc() -1 >>> QQ(Integer(42)).trunc() 42 >>> QQ(-Integer(42)).trunc() -42
(5/3).trunc() (-5/3).trunc() QQ(42).trunc() QQ(-42).trunc()
- val_unit(p)[source]¶
Return a pair: the \(p\)-adic valuation of
self, and the \(p\)-adic unit ofself, as aRational.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:
integer; the \(p\)-adic valuation of this rational
Rational; \(p\)-adic unit part ofself
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)
>>> from sage.all import * >>> (-Integer(4)/Integer(17)).val_unit(Integer(2)) (2, -1/17) >>> (-Integer(4)/Integer(17)).val_unit(Integer(17)) (-1, -4) >>> (Integer(0)/Integer(1)).val_unit(Integer(17)) (+Infinity, 1)
(-4/17).val_unit(2) (-4/17).val_unit(17) (0/1).val_unit(17)
AUTHORS:
David Roe (2007-04-12)
- valuation(p)[source]¶
Return the power of
pin the factorization ofself.INPUT:
p– a prime number
OUTPUT:
(integer or infinity)
Infinityifselfis zero, otherwise the (positive or negative) integer \(e\) such thatself= \(m*p^e\) with \(m\) coprime to \(p\).Note
See also
val_unit()which returns the pair \((e,m)\). The functionord()is an alias forvaluation().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
>>> from sage.all import * >>> x = -Integer(5)/Integer(9) >>> x.valuation(Integer(5)) 1 >>> x.ord(Integer(5)) 1 >>> x.valuation(Integer(3)) -2 >>> x.valuation(Integer(2)) 0
x = -5/9 x.valuation(5) x.ord(5) x.valuation(3) x.valuation(2)
Some edge cases:
sage: (0/1).valuation(4) +Infinity sage: (7/16).valuation(4) -2
>>> from sage.all import * >>> (Integer(0)/Integer(1)).valuation(Integer(4)) +Infinity >>> (Integer(7)/Integer(16)).valuation(Integer(4)) -2
(0/1).valuation(4) (7/16).valuation(4)
- class sage.rings.rational.Z_to_Q[source]¶
Bases:
MorphismA morphism from \(\ZZ\) to \(\QQ\).
- is_surjective()[source]¶
Return whether this morphism is surjective.
EXAMPLES:
sage: QQ.coerce_map_from(ZZ).is_surjective() False
>>> from sage.all import * >>> QQ.coerce_map_from(ZZ).is_surjective() False
QQ.coerce_map_from(ZZ).is_surjective()
- section()[source]¶
Return a section of this morphism.
EXAMPLES:
sage: f = QQ.coerce_map_from(ZZ).section(); f Generic map: From: Rational Field To: Integer Ring
>>> from sage.all import * >>> f = QQ.coerce_map_from(ZZ).section(); f Generic map: From: Rational Field To: Integer Ring
f = QQ.coerce_map_from(ZZ).section(); f
This map is a morphism in the category of sets with partial maps (see Issue #15618):
sage: f.parent() Set of Morphisms from Rational Field to Integer Ring in Category of sets with partial maps
>>> from sage.all import * >>> f.parent() Set of Morphisms from Rational Field to Integer Ring in Category of sets with partial maps
f.parent()
- class sage.rings.rational.int_to_Q[source]¶
Bases:
MorphismA morphism from Python 3
intto \(\QQ\).
- sage.rings.rational.integer_rational_power(a, b)[source]¶
Compute \(a^b\) as an integer, if it is integral, or return
None.The nonnegative real root is taken for even denominators.
INPUT:
a– anIntegerb– a nonnegativeRational
OUTPUT: \(a^b\) as an
IntegerorNoneEXAMPLES:
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
>>> from sage.all import * >>> from sage.rings.rational import integer_rational_power >>> integer_rational_power(Integer(49), Integer(1)/Integer(2)) 7 >>> integer_rational_power(Integer(27), Integer(1)/Integer(3)) 3 >>> integer_rational_power(-Integer(27), Integer(1)/Integer(3)) is None True >>> integer_rational_power(-Integer(27), Integer(2)/Integer(3)) is None True >>> integer_rational_power(Integer(512), Integer(7)/Integer(9)) 128 >>> integer_rational_power(Integer(27), Integer(1)/Integer(4)) is None True >>> integer_rational_power(-Integer(16), Integer(1)/Integer(4)) is None True >>> integer_rational_power(Integer(0), Integer(7)/Integer(9)) 0 >>> integer_rational_power(Integer(1), Integer(7)/Integer(9)) 1 >>> integer_rational_power(-Integer(1), Integer(7)/Integer(9)) is None True >>> integer_rational_power(-Integer(1), Integer(8)/Integer(9)) is None True >>> integer_rational_power(-Integer(1), Integer(9)/Integer(8)) is None True
from sage.rings.rational import integer_rational_power integer_rational_power(49, 1/2) integer_rational_power(27, 1/3) integer_rational_power(-27, 1/3) is None integer_rational_power(-27, 2/3) is None integer_rational_power(512, 7/9) integer_rational_power(27, 1/4) is None integer_rational_power(-16, 1/4) is None integer_rational_power(0, 7/9) integer_rational_power(1, 7/9) integer_rational_power(-1, 7/9) is None integer_rational_power(-1, 8/9) is None integer_rational_power(-1, 9/8) is None
TESTS (Issue #11228):
sage: integer_rational_power(-10, QQ(2)) 100 sage: integer_rational_power(0, QQ(0)) 1
>>> from sage.all import * >>> integer_rational_power(-Integer(10), QQ(Integer(2))) 100 >>> integer_rational_power(Integer(0), QQ(Integer(0))) 1
integer_rational_power(-10, QQ(2)) integer_rational_power(0, QQ(0))
- sage.rings.rational.is_Rational(x)[source]¶
Return
Trueifxis of the SageRationaltype.EXAMPLES:
sage: from sage.rings.rational import is_Rational sage: is_Rational(2) doctest:warning... DeprecationWarning: The function is_Rational is deprecated; use 'isinstance(..., Rational)' instead. See https://github.com/sagemath/sage/issues/38128 for details. False sage: is_Rational(2/1) True sage: is_Rational(int(2)) False sage: is_Rational('5') False
>>> from sage.all import * >>> from sage.rings.rational import is_Rational >>> is_Rational(Integer(2)) doctest:warning... DeprecationWarning: The function is_Rational is deprecated; use 'isinstance(..., Rational)' instead. See https://github.com/sagemath/sage/issues/38128 for details. False >>> is_Rational(Integer(2)/Integer(1)) True >>> is_Rational(int(Integer(2))) False >>> is_Rational('5') False
from sage.rings.rational import is_Rational is_Rational(2) is_Rational(2/1) is_Rational(int(2)) is_Rational('5')
- sage.rings.rational.make_rational(s)[source]¶
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
>>> from sage.all import * >>> (-Integer(7)/Integer(15)).str(Integer(32)) '-7/f' >>> sage.rings.rational.make_rational('-7/f') -7/15
(-7/15).str(32) sage.rings.rational.make_rational('-7/f')
- sage.rings.rational.rational_power_parts(a, b, factor_limit=100000)[source]¶
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 integerb– a rationalfactor_limit– the limit used in factoringa
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)
>>> from sage.all import * >>> from sage.rings.rational import rational_power_parts >>> rational_power_parts(Integer(27), Integer(1)/Integer(2)) (3, 3) >>> rational_power_parts(-Integer(128), Integer(3)/Integer(4)) (8, -8) >>> rational_power_parts(-Integer(4), Integer(1)/Integer(2)) (2, -1) >>> rational_power_parts(-Integer(4), Integer(1)/Integer(3)) (1, -4) >>> rational_power_parts(Integer(9)/Integer(1000), Integer(1)/Integer(2)) (3/10, 1/10)
from sage.rings.rational import rational_power_parts rational_power_parts(27, 1/2) rational_power_parts(-128, 3/4) rational_power_parts(-4, 1/2) rational_power_parts(-4, 1/3) rational_power_parts(9/1000, 1/2)