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.

TESTS:

sage: a = -2/3
sage: a == loads(dumps(a))
True

class sage.rings.rational.Q_to_Z¶

Bases: sage.categories.map.Map

TESTS:

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

section()¶

EXAMPLES:

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

class sage.rings.rational.Rational¶

Bases: sage.structure.element.FieldElement

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')
...
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

additive_order()¶

Return the additive order of self.

OUTPUT: integer or infinity

EXAMPLES:

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

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()
...
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

gcd(other, **kwds)¶

Return a gcd of the rational numbers self and other.

If self = other = 0, this is by convention 0. In all other cases it can (mathematically) be any nonzero rational number, but for simplicity we choose to always return 1.

EXAMPLES:

sage: gcd(1/3, 2/1)
1
sage: gcd(1/1, 0/1)
1
sage: gcd(0/1, 0/1)
0

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_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_nth_power(n)¶

Returns True if self is an nth 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

is_padic_square(p)¶

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

INPUT:

p - a prime number, or infinity

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
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 #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)¶

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')
 x - 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)
...
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 Q to Q 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 nth root of self, or raises a ValueError if self is not a perfect nth 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)
...
ValueError: not a perfect 2nd power

sage: (-25/4).nth_root(2)
...
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

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

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

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)
...
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

sqrt_approx(prec=None, all=False)¶

Return numerical approximation with given number of bits of precision to this rational number. If all is given, return both approximations.

INPUT:

prec - integer
all - bool

EXAMPLES:

sage: (5/3).sqrt_approx()
doctest:1172: DeprecationWarning: This function is deprecated.  Use sqrt with a given number of bits of precision instead.
1.29099444873581
sage: (990829038092384908234098239048230984/4).sqrt_approx()
4.9770197862083713747374920870362581922510725585130996993055116540856385e17
sage: (5/3).sqrt_approx(prec=200)
1.2909944487358056283930884665941332036109739017638636088625
sage: (9/4).sqrt_approx()
3/2

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)¶

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)
...
ValueError: base (=40) must be between 2 and 36
sage: (-4/17).str(1)
...
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()
...
ArithmeticError: Support of 0 not defined.

trace()¶

Returns the trace from Q to Q 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

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 $(e,m)$ .

EXAMPLES:

sage: x = -5/9
sage: x.valuation(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¶

Bases: sage.categories.morphism.Morphism

section()¶

EXAMPLES:

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

sage.rings.rational.clear_mpz_globals()¶

sage.rings.rational.gmp_randrange(n1, n2)¶

sage.rings.rational.init_mpz_globals()¶

class sage.rings.rational.int_to_Q¶: Bases: sage.categories.morphism.Morphism

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.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.pyrex_rational_reconstruction(a, m)¶

Find the rational reconstruction of a mod m, if it exists.

INPUT:

a - Integer
m - Integer

OUTPUT:

x - rings.rational.Rational

EXAMPLES:

sage: Integers(100)(2/3)
34
sage: sage.rings.rational.pyrex_rational_reconstruction(34, 100)
2/3

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 #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

Rational Numbers¶

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