Module: sage.rings.rational
Author Log:
Module-level Functions
) |
s) |
x) |
Output x formated for inclusion in a MathML document.
integer.Integer a, integer.Integer m) |
Find the rational reconstruction of a mod m, if it exists.
INPUT: a -- Integer m -- Integer OUTPUT: x -- rings.rational.Rational
Class: Rational
Functions: additive_order,
ceil,
copy,
denom,
denominator,
factor,
floor,
gcd,
height,
is_one,
is_zero,
lcm,
mod_ui,
multiplicative_order,
numer,
numerator,
parent,
set_si,
set_str,
sqrt,
square_root,
str,
valuation
self) |
Return the additive order of self.
sage: QQ(0).additive_order() 1 sage: QQ(1).additive_order() Infinity
self) |
self.ceil(): Return the ceiling of this rational number.
If this rational number is an integer, this returns this number, otherwise it returns the floor of this number +1.
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
self) |
self.denom(): Return the denominator of this rational number.
sage: x = 5/13 sage: x.denom() 13 sage: x = -9/3 sage: x.denom() 1
self) |
self.denominator(): Return the denominator of this rational number.
sage: x = -5/11 sage: x.denominator() 11 sage: x = 9/3 sage: x.denominator() 1
self) |
self.floor(): Return the floor of this rational number.
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
self, Rational other) |
Return the least common multiple of self and other.
Our hopefully interesting notion of GCD for rational numbers is illustrated in the examples below.
sage: gcd(2/3,1/5) 1/15 sage: gcd(2/3,7/5) 1/15 sage: gcd(1/3,1/6) 1/6 sage: gcd(6/7,9/7) 3/7
self) |
The max absolute value of the numerator and denominator of self, as an Integer.
sage: a = 2/3 sage: a.height() 3 sage: a = 34/3 sage: a.height() 34 sage: a = -97/4 sage: a.height() 97
Author: Naqi Jaffery (2006-03-05): examples
self, Rational other) |
Return the least common multiple of self and other.
Our hopefully interesting notion of LCM for rational numbers is illustrated in the examples below.
sage: lcm(2/3,1/5) 2 sage: lcm(2/3,7/5) 14 sage: lcm(1/3,1/5) 1 sage: lcm(1/3,1/6) 1/3
self) |
Return the multiplicative order of self, if self is a unit, or raise codeArithmeticError otherwise.
sage: QQ(1).multiplicative_order() 1 sage: QQ('1/-1').multiplicative_order() 2 sage: QQ(0).multiplicative_order() Traceback (most recent call last): ... ArithmeticError: no power of 0 is a unit sage: QQ('2/3').multiplicative_order() Traceback (most recent call last): ... ArithmeticError: no power of 2/3 is a unit
self) |
Return the numerator of this rational number.
sage: x = -5/11 sage: x.numer() -5
self) |
Return the numerator of this rational number.
sage: x = 5/11 sage: x.numerator() 5
sage: x = 9/3 sage: x.numerator() 3
self, bits=None) |
Returns the positive square root of self as a real number to the given number of bits of precision if self is nonnegative, and raises a ValueError exception otherwise.
INPUT: bits -- number of bits of precision. If bits is not specified, the number of bits of precision is at least twice the number of bits of self (the precision is always at least 53 bits if not specified). OUTPUT: integer, real number, or complex number.
sage: x = 23/2 sage: x.sqrt() 3.3911649915626341 sage: x = 32/5 sage: x.sqrt() 2.5298221281347035 sage: x = 16/9 sage: x.sqrt() 4/3 sage: x.sqrt(53) 1.3333333333333333 sage: x = 9837/2 sage: x.sqrt() 70.132018365365752 sage: x = 645373/45 sage: x.sqrt() 119.75651223303984 sage: x = -12/5 sage: x.sqrt() 1.5491933384829668*I
Author: Naqi Jaffery (2006-03-05): examples
self) |
Return the positive rational square root of self, or raises a ValueError if self is not a perfect square.
sage: x = 125/5 sage: x.square_root() 5 sage: x = 64/4 sage: x.square_root() 4 sage: x = 1000/10 sage: x.square_root() 10 sage: x = 81/3 sage: x.square_root() Traceback (most recent call last): ... ValueError: self (=27) is not a perfect square
Author: Naqi Jaffery (2006-03-05): examples
Special Functions: __abs__,
__add_,
__add__,
__cmp__,
__div_,
__div__,
__eq__,
__float__,
__ge__,
__getitem__,
__gt__,
__int__,
__invert__,
__le__,
__long__,
__lshift__,
__lt__,
__mod__,
__mul_,
__mul__,
__ne__,
__neg__,
__pos__,
__pow__,
__radd__,
__rdiv__,
__reduce__,
__repr__,
__rlshift__,
__rmod__,
__rmul__,
__rpow__,
__rrshift__,
__rshift__,
__rsub__,
__set_value,
__sub_,
__sub__,
_gcd,
_im_gens_,
_interface_init_,
_latex_,
_lcm,
_lshift,
_mathml_,
_mpfr_,
_pari_,
_reduce_set,
_rshift
self, x) |
sage: a = long(901824309821093821093812093810928309183091832091) sage: b = QQ(a); b 901824309821093821093812093810928309183091832091 sage: QQ(b) 901824309821093821093812093810928309183091832091 sage: QQ(int(93820984323)) 93820984323 sage: QQ(ZZ(901824309821093821093812093810928309183091832091)) 901824309821093821093812093810928309183091832091 sage: QQ('-930482/9320842317') -930482/9320842317 sage: QQ((-930482, 9320842317)) -930482/9320842317 sage: QQ([9320842317]) 9320842317 sage: QQ(pari(39029384023840928309482842098430284398243982394)) 39029384023840928309482842098430284398243982394 sage: QQ('sage') Traceback (most recent call last): ... TypeError: unable to convert sage to a rational
sage: QQ(RR(3929329/32)) 3929329/32 sage: QQ(RR(1/7)) - 1/7 -1/126100789566373888 sage: QQ(23.2) 6530219459687219/281474976710656 sage: 6530219459687219.0/281474976710656 23.1999999999999993
self, Rational other) |
Returns the least common multiple, in the rational numbers, of self and other. This function returns either 0 or 1 (as a rational number).
self) |
sage: kash(3/1).Type() # optional elt-fld^rat sage: magma(3/1).Type() # optional FldRatElt
self, Rational other) |
Returns the least common multiple, in the rational numbers, of self and other. This function returns either 0 or 1 (as a rational number).
self, unsigned long int exp) |
Return
self, unsigned long int exp) |
Return
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