ďż˝ !A?hpďż˝ďż˝ďż˝ďż˝dZddlmZddlZddlZddlZddlZddlZdgZej j Zej jZ ejdejejzďż˝ďż˝ZGdďż˝dejďż˝ďż˝ZdS)z/Fraction, infinite-precision, rational numbers.ďż˝ďż˝ďż˝DecimalNďż˝Fractionaďż˝ \A\s* # optional whitespace at the start, (?P[-+]?) # an optional sign, then (?=\d|\.\d) # lookahead for digit or .digit (?P\d*|\d+(_\d+)*) # numerator (possibly empty) (?: # followed by (?:/(?P\d+(_\d+)*))? # an optional denominator | # or (?:\.(?P\d*|\d+(_\d+)*))? # an optional fractional part (?:E(?P[-+]?\d+(_\d+)*))? # and optional exponent ) \s*\Z # and optional whitespace to finish cďż˝lďż˝ďż˝eZdZdZdZd.ddďż˝ďż˝fdďż˝Zedďż˝ďż˝ďż˝Zed ďż˝ďż˝ďż˝Zd ďż˝Z d/dďż˝Z ed ďż˝ďż˝ďż˝Zedďż˝ďż˝ďż˝Z dďż˝Zdďż˝Zdďż˝Zdďż˝Zeeejďż˝ďż˝\ZZdďż˝Zeeejďż˝ďż˝\ZZdďż˝Zeeejďż˝ďż˝\ZZdďż˝Zeeejďż˝ďż˝\Z Z!dďż˝Z"ee"ej#ďż˝ďż˝\Z$Z%dďż˝Z&ee&e'ďż˝ďż˝\Z(Z)dďż˝Z*ee*ej+ďż˝ďż˝\Z,Z-dďż˝Z.dďż˝Z/dďż˝Z0dďż˝Z1dďż˝Z2ej3fdďż˝Z4dďż˝Z5d ďż˝Z6d!ďż˝Z7d0d"ďż˝Z8d#ďż˝Z9d$ďż˝Z:d%ďż˝Z;d&ďż˝Zd)ďż˝Z?d*ďż˝Z@d+ďż˝ZAd,ďż˝ZBd-ďż˝ZCďż˝xZDS)1ra]This class implements rational numbers. In the two-argument form of the constructor, Fraction(8, 6) will produce a rational number equivalent to 4/3. Both arguments must be Rational. The numerator defaults to 0 and the denominator defaults to 1 so that Fraction(3) == 3 and Fraction() == 0. Fractions can also be constructed from: - numeric strings similar to those accepted by the float constructor (for example, '-2.3' or '1e10') - strings of the form '123/456' - float and Decimal instances - other Rational instances (including integers) ďż˝ďż˝ _numeratorďż˝_denominatorrNTďż˝ďż˝ _normalizecďż˝ďż˝ďż˝ďż˝tt|ďż˝ďż˝ďż˝|ďż˝ďż˝}|ďż˝ďż˝ďż˝t|ďż˝ďż˝tur||_d|_|St|tj ďż˝ďż˝r|j |_|j|_|St|ttfďż˝ďż˝r#|ďż˝ďż˝ďż˝\|_|_|St|tďż˝ďż˝ďż˝r/t ďż˝|ďż˝ďż˝}|ďż˝t%d|zďż˝ďż˝ďż˝t |ďż˝dďż˝ďż˝pdďż˝ďż˝}|ďż˝dďż˝ďż˝}|rt |ďż˝ďż˝}nďż˝d}|ďż˝dďż˝ďż˝}|rB|ďż˝dd ďż˝ďż˝}d t+|ďż˝ďż˝z}||zt |ďż˝ďż˝z}||z}|ďż˝dďż˝ďż˝} | r't | ďż˝ďż˝} | dkr |d | zz}n |d | zz}|ďż˝d ďż˝ďż˝dkr|}nďż˝t-dďż˝ďż˝ďż˝t|ďż˝ďż˝tcxurt|ďż˝ďż˝urnnnbt|tj ďż˝ďż˝r9t|tj ďż˝ďż˝r|j |jz|j |jz}}nt-dďż˝ďż˝ďż˝|dkrt/d|zďż˝ďż˝ďż˝|r(t1j||ďż˝ďż˝} |dkr| } || z}|| z}||_||_|S)aďż˝Constructs a Rational. Takes a string like '3/2' or '1.5', another Rational instance, a numerator/denominator pair, or a float. Examples -------- >>> Fraction(10, -8) Fraction(-5, 4) >>> Fraction(Fraction(1, 7), 5) Fraction(1, 35) >>> Fraction(Fraction(1, 7), Fraction(2, 3)) Fraction(3, 14) >>> Fraction('314') Fraction(314, 1) >>> Fraction('-35/4') Fraction(-35, 4) >>> Fraction('3.1415') # conversion from numeric string Fraction(6283, 2000) >>> Fraction('-47e-2') # string may include a decimal exponent Fraction(-47, 100) >>> Fraction(1.47) # direct construction from float (exact conversion) Fraction(6620291452234629, 4503599627370496) >>> Fraction(2.25) Fraction(9, 4) >>> Fraction(Decimal('1.47')) Fraction(147, 100) Nďż˝z Invalid literal for Fraction: %rďż˝numďż˝0ďż˝denomďż˝decimalďż˝_ďż˝ďż˝ ďż˝exprďż˝signďż˝-z2argument should be a string or a Rational instancez+both arguments should be Rational instanceszFraction(%s, 0))ďż˝superrďż˝__new__ďż˝typeďż˝intrr ďż˝ isinstanceďż˝numbersďż˝Rationalďż˝ numeratorďż˝denominatorďż˝floatrďż˝as_integer_ratioďż˝strďż˝_RATIONAL_FORMATďż˝matchďż˝ ValueErrorďż˝groupďż˝replaceďż˝lenďż˝ TypeErrorďż˝ZeroDivisionErrorďż˝mathďż˝gcd)ďż˝clsrr rďż˝selfďż˝mrrďż˝scalerďż˝gďż˝ __class__s ďż˝ďż˝6/opt/alt/python-internal/lib64/python3.11/fractions.pyrzFraction.__new__>s-ďż˝ďż˝ďż˝>ďż˝Xďż˝sďż˝#ďż˝#ďż˝+ďż˝+ďż˝Cďż˝0ďż˝0ďż˝ďż˝ďż˝ďż˝ďż˝Iďż˝ďż˝ďż˝#ďż˝%ďż˝%ďż˝"+ďż˝ďż˝ďż˝$%ďż˝ďż˝!ďż˝ďż˝ďż˝ďż˝Iďż˝wďż˝'7ďż˝8ďż˝8ďż˝( :ďż˝"+ďż˝"5ďż˝ďż˝ďż˝$-ďż˝$9ďż˝ďż˝!ďż˝ďż˝ďż˝ďż˝Iďż˝ďż˝wďż˝'7ďż˝8ďż˝8ďż˝# :ďż˝5>ďż˝5Oďż˝5Oďż˝5Qďż˝5Qďż˝2ďż˝ďż˝ďż˝ďż˝!2ďż˝ďż˝ďż˝ďż˝Iďż˝sďż˝+ďż˝+ďż˝ :ďż˝$ďż˝*ďż˝*ďż˝9ďż˝5ďż˝5ďż˝ďż˝ďż˝9ďż˝$ďż˝%Gďż˝%.ďż˝&/ďż˝0ďż˝0ďż˝0ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ 5ďż˝#ďż˝6ďż˝6ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝(ďż˝(ďż˝ďż˝ďż˝4ďż˝"%ďż˝eďż˝*ďż˝*ďż˝Kďż˝Kďż˝"#ďż˝Kďż˝ďż˝gďż˝gďż˝iďż˝0ďż˝0ďż˝Gďż˝ďż˝-ďż˝")ďż˝/ďż˝/ďż˝#ďż˝rďż˝":ďż˝":ďż˝ďż˝ "ďż˝Cďż˝ďż˝Lďż˝Lďż˝ 0ďż˝ďż˝$-ďż˝ďż˝$5ďż˝ďż˝Gďż˝ďż˝ďż˝$Dďż˝ ďż˝#ďż˝uďż˝,ďż˝ďż˝ďż˝'ďż˝'ďż˝%ďż˝.ďż˝.ďż˝Cďż˝ďż˝4ďż˝!ďż˝#ďż˝hďż˝hďż˝ďż˝ďż˝!ďż˝8ďż˝8ďż˝%ďż˝ďż˝Sďż˝ďż˝0ďż˝Iďż˝Iďż˝'ďż˝2ďż˝ďż˝tďż˝8ďż˝3ďż˝Kďż˝ďż˝7ďż˝7ďż˝6ďż˝?ďż˝?ďż˝cďż˝)ďż˝)ďż˝!*ďż˝ ďż˝Iďż˝ďż˝ ďż˝!9ďż˝:ďż˝:ďż˝:ďż˝ďż˝)ďż˝_ďż˝_ďż˝ďż˝ 8ďż˝ 8ďż˝ 8ďż˝ 8ďż˝tďż˝Kďż˝'8ďż˝'8ďż˝ 8ďż˝ 8ďż˝ 8ďż˝ 8ďż˝ 8ďż˝ďż˝ďż˝ďż˝Gďż˝$4ďż˝5ďż˝5ďż˝ 2ďż˝ďż˝{ďż˝Gďż˝$4ďż˝5ďż˝5ďż˝ 2ďż˝ďż˝#ďż˝kďż˝&=ďż˝=ďż˝ďż˝%ďż˝ ďż˝(=ďż˝=ďż˝#ďż˝Iďż˝Iďż˝ ďż˝1ďż˝2ďż˝2ďż˝ 2ďż˝ďż˝!ďż˝ďż˝ďż˝#ďż˝$5ďż˝ ďż˝$Aďż˝Bďż˝Bďż˝Bďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝Kďż˝0ďż˝0ďż˝Aďż˝ďż˝Qďż˝ďż˝ďż˝ďż˝Bďż˝ďż˝ďż˝!ďż˝Oďż˝Iďż˝ďż˝Aďż˝ďż˝Kďż˝#ďż˝ďż˝ďż˝'ďż˝ďż˝ďż˝ďż˝ďż˝c ďż˝ďż˝t|tjďż˝ďż˝r||ďż˝ďż˝St|tďż˝ďż˝s/t |jďż˝d|ďż˝dt |ďż˝ďż˝jďż˝dďż˝ďż˝ďż˝ďż˝||ďż˝ďż˝ďż˝ďż˝S)zďż˝Converts a finite float to a rational number, exactly. Beware that Fraction.from_float(0.3) != Fraction(3, 10). z%.from_float() only takes floats, not ďż˝ (ďż˝))rrďż˝Integralr!r*ďż˝__name__rr")r.ďż˝fs r4ďż˝ from_floatzFraction.from_floatďż˝sďż˝ďż˝ďż˝ďż˝aďż˝ďż˝)ďż˝*ďż˝*ďż˝ Aďż˝ďż˝3ďż˝qďż˝6ďż˝6ďż˝Mďż˝ďż˝Aďż˝uďż˝%ďż˝%ďż˝ Aďż˝ďż˝ ďż˝\ďż˝\ďż˝\ďż˝1ďż˝1ďż˝1ďż˝dďż˝1ďż˝gďż˝gďż˝.>ďż˝.>ďż˝.>ďż˝@ďż˝Aďż˝Aďż˝ Aďż˝ďż˝sďż˝Aďż˝&ďż˝&ďż˝(ďż˝(ďż˝)ďż˝)r5c ďż˝ ďż˝ddlm}t|tjďż˝ďż˝r|t|ďż˝ďż˝ďż˝ďż˝}n?t||ďż˝ďż˝s/t |jďż˝d|ďż˝dt|ďż˝ďż˝jďż˝dďż˝ďż˝ďż˝ďż˝||ďż˝ ďż˝ďż˝ďż˝S)zAConverts a finite Decimal instance to a rational number, exactly.rrz).from_decimal() only takes Decimals, not r7r8) rrrrr9rr*r:rr")r.ďż˝decrs r4ďż˝from_decimalzFraction.from_decimalďż˝sďż˝ďż˝ďż˝ $ďż˝#ďż˝#ďż˝#ďż˝#ďż˝#ďż˝ďż˝cďż˝7ďż˝+ďż˝,ďż˝,ďż˝ 9ďż˝ďż˝'ďż˝#ďż˝cďż˝(ďż˝(ďż˝#ďż˝#ďż˝Cďż˝Cďż˝ďż˝Cďż˝ďż˝)ďż˝)ďż˝ 9ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝sďż˝sďż˝sďż˝Dďż˝ďż˝Iďż˝Iďż˝$6ďż˝$6ďż˝$6ďż˝8ďż˝9ďż˝9ďż˝ 9ďż˝ďż˝sďż˝Cďż˝(ďż˝(ďż˝*ďż˝*ďż˝+ďż˝+r5cďż˝ďż˝|j|jfS)zďż˝Return the integer ratio as a tuple. Return a tuple of two integers, whose ratio is equal to the Fraction and with a positive denominator. rďż˝r/s r4r"zFraction.as_integer_ratioďż˝sďż˝ďż˝ďż˝ďż˝ďż˝!2ďż˝3ďż˝3r5ďż˝@Bcďż˝ďż˝|dkrtdďż˝ďż˝ďż˝|j|krt|ďż˝ďż˝Sd\}}}}|j|j}} ||z}|||zz} | |krn|||||zz| f\}}}}||||zz }}ďż˝0||z |z} t|| |zz|| |zzďż˝ďż˝}t||ďż˝ďż˝}t ||z ďż˝ďż˝t ||z ďż˝ďż˝kr|S|S)aWClosest Fraction to self with denominator at most max_denominator. >>> Fraction('3.141592653589793').limit_denominator(10) Fraction(22, 7) >>> Fraction('3.141592653589793').limit_denominator(100) Fraction(311, 99) >>> Fraction(4321, 8765).limit_denominator(10000) Fraction(4321, 8765) r z$max_denominator should be at least 1)rr r r)r&r rrďż˝abs) r/ďż˝max_denominatorďż˝p0ďż˝q0ďż˝p1ďż˝q1ďż˝nďż˝dďż˝aďż˝q2ďż˝kďż˝bound1ďż˝bound2s r4ďż˝limit_denominatorzFraction.limit_denominatorďż˝sďż˝ďż˝@ďż˝Qďż˝ďż˝ďż˝ďż˝Cďż˝Dďż˝Dďż˝Dďż˝ďż˝ďż˝ďż˝/ďż˝/ďż˝ďż˝Dďż˝>ďż˝>ďż˝!ďż˝#ďż˝ďż˝ďż˝Bďż˝ďż˝Bďż˝ďż˝ďż˝ďż˝ 1ďż˝1ďż˝ďż˝ ďż˝ďż˝1ďż˝ďż˝Aďż˝ďż˝Aďż˝bďż˝Dďż˝ďż˝Bďż˝ďż˝Oďż˝#ďż˝#ďż˝ďż˝ďż˝ďż˝Rďż˝ďż˝"ďż˝ďż˝Wďż˝bďż˝0ďż˝Nďż˝Bďż˝ďż˝Bďż˝ďż˝ďż˝aďż˝ďż˝!ďż˝ďż˝eďż˝qďż˝Aďż˝ ďż˝ďż˝Rďż˝ ďż˝"ďż˝$ďż˝ďż˝ďż˝"ďż˝Qďż˝rďż˝Tďż˝'ďż˝2ďż˝aďż˝ďż˝dďż˝7ďż˝+ďż˝+ďż˝ďż˝ďż˝"ďż˝bďż˝!ďż˝!ďż˝ďż˝ďż˝vďż˝ďż˝}ďż˝ďż˝ďż˝ďż˝Vďż˝Dďż˝[ďż˝!1ďż˝!1ďż˝1ďż˝1ďż˝ďż˝Mďż˝ďż˝Mr5cďż˝ďż˝|jSďż˝N)rďż˝rLs r4rzFraction.numerators ďż˝ďż˝ďż˝|ďż˝r5cďż˝ďż˝|jSrS)r rTs r4r zFraction.denominators ďż˝ďż˝ďż˝~ďż˝r5cďż˝@ďż˝|jjďż˝d|jďż˝d|jďż˝dďż˝S)z repr(self)ďż˝(z, r8)r3r:rr rAs r4ďż˝__repr__zFraction.__repr__ s0ďż˝ďż˝#ďż˝~ďż˝6ďż˝6ďż˝6ďż˝#ďż˝ďż˝ďż˝ďż˝ďż˝0Aďż˝0Aďż˝0Aďż˝Cďż˝ Cr5cďż˝bďż˝|jdkrt|jďż˝ďż˝S|jďż˝d|jďż˝ďż˝S)z str(self)r ďż˝/)r r#rrAs r4ďż˝__str__zFraction.__str__s7ďż˝ďż˝ďż˝ďż˝ďż˝!ďż˝!ďż˝ďż˝tďż˝ďż˝'ďż˝'ďż˝'ďż˝"ďż˝oďż˝oďż˝oďż˝tďż˝/@ďż˝/@ďż˝Aďż˝Ar5cďż˝ďż˝ďż˝ďż˝ďż˝ďż˝fdďż˝}dďż˝jzdz|_ďż˝j|_ďż˝ďż˝fdďż˝}dďż˝jzdz|_ďż˝j|_||fS)aďż˝Generates forward and reverse operators given a purely-rational operator and a function from the operator module. Use this like: __op__, __rop__ = _operator_fallbacks(just_rational_op, operator.op) In general, we want to implement the arithmetic operations so that mixed-mode operations either call an implementation whose author knew about the types of both arguments, or convert both to the nearest built in type and do the operation there. In Fraction, that means that we define __add__ and __radd__ as: def __add__(self, other): # Both types have numerators/denominator attributes, # so do the operation directly if isinstance(other, (int, Fraction)): return Fraction(self.numerator * other.denominator + other.numerator * self.denominator, self.denominator * other.denominator) # float and complex don't have those operations, but we # know about those types, so special case them. elif isinstance(other, float): return float(self) + other elif isinstance(other, complex): return complex(self) + other # Let the other type take over. return NotImplemented def __radd__(self, other): # radd handles more types than add because there's # nothing left to fall back to. if isinstance(other, numbers.Rational): return Fraction(self.numerator * other.denominator + other.numerator * self.denominator, self.denominator * other.denominator) elif isinstance(other, Real): return float(other) + float(self) elif isinstance(other, Complex): return complex(other) + complex(self) return NotImplemented There are 5 different cases for a mixed-type addition on Fraction. I'll refer to all of the above code that doesn't refer to Fraction, float, or complex as "boilerplate". 'r' will be an instance of Fraction, which is a subtype of Rational (r : Fraction <: Rational), and b : B <: Complex. The first three involve 'r + b': 1. If B <: Fraction, int, float, or complex, we handle that specially, and all is well. 2. If Fraction falls back to the boilerplate code, and it were to return a value from __add__, we'd miss the possibility that B defines a more intelligent __radd__, so the boilerplate should return NotImplemented from __add__. In particular, we don't handle Rational here, even though we could get an exact answer, in case the other type wants to do something special. 3. If B <: Fraction, Python tries B.__radd__ before Fraction.__add__. This is ok, because it was implemented with knowledge of Fraction, so it can handle those instances before delegating to Real or Complex. The next two situations describe 'b + r'. We assume that b didn't know about Fraction in its implementation, and that it uses similar boilerplate code: 4. If B <: Rational, then __radd_ converts both to the builtin rational type (hey look, that's us) and proceeds. 5. Otherwise, __radd__ tries to find the nearest common base ABC, and fall back to its builtin type. Since this class doesn't subclass a concrete type, there's no implementation to fall back to, so we need to try as hard as possible to return an actual value, or the user will get a TypeError. cďż˝ďż˝ďż˝t|ttfďż˝ďż˝rďż˝||ďż˝ďż˝St|tďż˝ďż˝rďż˝t|ďż˝ďż˝|ďż˝ďż˝St|tďż˝ďż˝rďż˝t |ďż˝ďż˝|ďż˝ďż˝St SrS)rrrr!ďż˝complexďż˝NotImplemented)rLďż˝bďż˝fallback_operatorďż˝monomorphic_operators ďż˝ďż˝r4ďż˝forwardz-Fraction._operator_fallbacks..forwardesďż˝ďż˝ďż˝ďż˝ďż˝!ďż˝cďż˝8ďż˝_ďż˝-ďż˝-ďż˝ &ďż˝+ďż˝+ďż˝Aďż˝qďż˝1ďż˝1ďż˝1ďż˝ďż˝Aďż˝uďż˝%ďż˝%ďż˝ &ďż˝(ďż˝(ďż˝ďż˝qďż˝ďż˝ďż˝1ďż˝5ďż˝5ďż˝5ďż˝ďż˝Aďż˝wďż˝'ďż˝'ďż˝ &ďż˝(ďż˝(ďż˝ďż˝ďż˝ďż˝ďż˝Qďż˝7ďż˝7ďż˝7ďż˝%ďż˝%r5ďż˝__cďż˝^ďż˝ďż˝t|tjďż˝ďż˝rďż˝||ďż˝ďż˝St|tjďż˝ďż˝r&ďż˝t |ďż˝ďż˝t |ďż˝ďż˝ďż˝ďż˝St|tjďż˝ďż˝r&ďż˝t |ďż˝ďż˝t |ďż˝ďż˝ďż˝ďż˝StSrS)rrrďż˝Realr!ďż˝Complexr^r_)r`rLrarbs ďż˝ďż˝r4ďż˝reversez-Fraction._operator_fallbacks..reverseqsďż˝ďż˝ďż˝ďż˝ďż˝!ďż˝Wďż˝-ďż˝.ďż˝.ďż˝ &ďż˝+ďż˝+ďż˝Aďż˝qďż˝1ďż˝1ďż˝1ďż˝ďż˝Aďż˝wďż˝|ďż˝,ďż˝,ďż˝ &ďż˝(ďż˝(ďż˝ďż˝qďż˝ďż˝ďż˝5ďż˝ďż˝8ďż˝8ďż˝<ďż˝<ďż˝<ďż˝ďż˝Aďż˝wďż˝ďż˝/ďż˝/ďż˝ &ďż˝(ďż˝(ďż˝ďż˝ďż˝ďż˝ďż˝Wďż˝Qďż˝Zďż˝Zďż˝@ďż˝@ďż˝@ďż˝%ďż˝%r5ďż˝__r)r:ďż˝__doc__)rbrarcrhs`` r4ďż˝_operator_fallbackszFraction._operator_fallbackssďż˝ďż˝ďż˝ďż˝ďż˝` &ďż˝ &ďż˝ &ďż˝ &ďż˝ &ďż˝ &ďż˝ ďż˝"3ďż˝"<ďż˝<ďż˝tďż˝Cďż˝ďż˝ďż˝.ďż˝6ďż˝ďż˝ďż˝ &ďż˝ &ďż˝ &ďż˝ &ďż˝ &ďż˝ &ďż˝!ďż˝#4ďż˝#=ďż˝=ďż˝ďż˝Dďż˝ďż˝ďż˝.ďż˝6ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝r5cďż˝hďż˝|j|j}}|j|j}}tj||ďż˝ďż˝}|dkrt ||z||zz||zdďż˝ďż˝ďż˝S||z}|||zz||zz}tj||ďż˝ďż˝} | dkrt |||zdďż˝ďż˝ďż˝St || z||| zzdďż˝ďż˝ďż˝S)za + br Fr ďż˝rr r,r-rďż˝ rLr`ďż˝naďż˝daďż˝nbďż˝dbr2ďż˝sďż˝tďż˝g2s r4ďż˝_addz Fraction._addďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝aďż˝mďż˝Bďż˝ďż˝ďż˝ďż˝aďż˝mďż˝Bďż˝ďż˝ďż˝Hďż˝Rďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝6ďż˝6ďż˝ďż˝Bďż˝ďż˝Gďż˝bďż˝2ďż˝gďż˝-ďż˝rďż˝Bďż˝wďż˝5ďż˝Iďż˝Iďż˝Iďż˝Iďż˝ďż˝!ďż˝Gďż˝ďż˝ďż˝"ďż˝ďż˝'ďż˝Nďż˝Rďż˝!ďż˝Vďż˝#ďż˝ďż˝ ďż˝Xďż˝aďż˝ďż˝^ďż˝^ďż˝ďż˝ ďż˝ďż˝7ďż˝7ďż˝ďż˝Aďż˝qďż˝2ďż˝vďż˝%ďż˝8ďż˝8ďż˝8ďż˝8ďż˝ďż˝ďż˝Rďż˝ďż˝ďż˝bďż˝Bďż˝hďż˝ďż˝Eďż˝Bďż˝Bďż˝Bďż˝Br5cďż˝hďż˝|j|j}}|j|j}}tj||ďż˝ďż˝}|dkrt ||z||zz ||zdďż˝ďż˝ďż˝S||z}|||zz||zz }tj||ďż˝ďż˝} | dkrt |||zdďż˝ďż˝ďż˝St || z||| zzdďż˝ďż˝ďż˝S)za - br Fr rmrns r4ďż˝_subz Fraction._subďż˝rwr5cďż˝ďż˝ďż˝|j|j}}|j|j}}tj||ďż˝ďż˝}|dkr ||z}||z}tj||ďż˝ďż˝}|dkr ||z}||z}t ||z||zdďż˝ďż˝ďż˝S)za * br Fr rm)rLr`rorprqrrďż˝g1rus r4ďż˝_mulz Fraction._mulďż˝sďż˝ďż˝ďż˝ďż˝ďż˝aďż˝mďż˝Bďż˝ďż˝ďż˝ďż˝aďż˝mďż˝Bďż˝ďż˝ ďż˝Xďż˝bďż˝"ďż˝ ďż˝ ďż˝ďż˝ ďż˝ďż˝6ďż˝6ďż˝ďż˝2ďż˝Iďż˝Bďż˝ďż˝2ďż˝Iďż˝Bďż˝ ďż˝Xďż˝bďż˝"ďż˝ ďż˝ ďż˝ďż˝ ďż˝ďż˝6ďż˝6ďż˝ďż˝2ďż˝Iďż˝Bďż˝ďż˝2ďż˝Iďż˝Bďż˝ďż˝ďż˝Rďż˝ďż˝ďż˝bďż˝ďż˝Uďż˝;ďż˝;ďż˝;ďż˝;r5cďż˝ďż˝|j|j}}|j|j}}tj||ďż˝ďż˝}|dkr ||z}||z}tj||ďż˝ďż˝}|dkr ||z}||z}||z||z} }| dkr|| } }t || dďż˝ďż˝ďż˝S)za / br rFr rm) rLr`rorprqrrr{rurJrKs r4ďż˝_divz Fraction._divďż˝sďż˝ďż˝ďż˝ďż˝ďż˝aďż˝mďż˝Bďż˝ďż˝ďż˝ďż˝aďż˝mďż˝Bďż˝ďż˝ ďż˝Xďż˝bďż˝"ďż˝ ďż˝ ďż˝ďż˝ ďż˝ďż˝6ďż˝6ďż˝ďż˝2ďż˝Iďż˝Bďż˝ďż˝2ďż˝Iďż˝Bďż˝ ďż˝Xďż˝bďż˝"ďż˝ ďż˝ ďż˝ďż˝ ďż˝ďż˝6ďż˝6ďż˝ďż˝2ďż˝Iďż˝Bďż˝ďż˝2ďż˝Iďż˝Bďż˝ďż˝Bďż˝wďż˝ďż˝Rďż˝ďż˝1ďż˝ďż˝ďż˝qďż˝5ďż˝5ďż˝ďż˝2ďż˝ďż˝rďż˝qďż˝Aďż˝ďż˝ďż˝1ďż˝ďż˝/ďż˝/ďż˝/ďż˝/r5cďż˝@ďż˝|j|jz|j|jzzS)za // bďż˝rr ďż˝rLr`s r4ďż˝ _floordivzFraction._floordivsďż˝ďż˝ďż˝ďż˝aďż˝mďż˝+ďż˝ďż˝ďż˝ďż˝ďż˝1Lďż˝Mďż˝Mr5cďż˝ďż˝|j|j}}t|j|z||jzďż˝ďż˝\}}|t|||zďż˝ďż˝fS)z(a // b, a % b))r ďż˝divmodrr)rLr`rprrďż˝divďż˝n_mods r4ďż˝_divmodzFraction._divmodsIďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝Bďż˝ďż˝ďż˝Aďż˝Kďż˝"ďż˝,ďż˝bďż˝1ďż˝;ďż˝.>ďż˝?ďż˝?ďż˝ ďż˝ďż˝Uďż˝ďż˝Hďż˝Uďż˝Bďż˝ďż˝Gďż˝,ďż˝,ďż˝,ďż˝,r5cďż˝jďż˝|j|j}}t|j|z|j|zz||zďż˝ďż˝S)za % b)r rr)rLr`rprrs r4ďż˝_modz Fraction._mods6ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝Bďż˝ďż˝ďż˝ďż˝ďż˝rďż˝)ďż˝aďż˝kďż˝Bďż˝.>ďż˝?ďż˝ďż˝bďż˝ďż˝Iďż˝Iďż˝Ir5cďż˝ďż˝t|tjďż˝ďż˝rďż˝|jdkrďż˝|j}|dkr"t|j|z|j|zdďż˝ďż˝ďż˝S|jdkr$t|j|z|j|zdďż˝ďż˝ďż˝St|j|z|j|zdďż˝ďż˝ďż˝St|ďż˝ďż˝t|ďż˝ďż˝zSt|ďż˝ďż˝|zS)zďż˝a ** b If b is not an integer, the result will be a float or complex since roots are generally irrational. If b is an integer, the result will be rational. r rFr ) rrrr rrrr r!)rLr`ďż˝powers r4ďż˝__pow__zFraction.__pow__sďż˝ďż˝ďż˝aďż˝ďż˝)ďż˝*ďż˝*ďż˝ !ďż˝ďż˝}ďż˝ďż˝!ďż˝!ďż˝ďż˝ďż˝ďż˝ďż˝Aďż˝:ďż˝:ďż˝#ďż˝Aďż˝Lďż˝Eďż˝$9ďż˝$%ďż˝Nďż˝eďż˝$;ďż˝/4ďż˝6ďż˝6ďż˝6ďż˝6ďż˝ďż˝\ďż˝Qďż˝&ďż˝&ďż˝#ďż˝Aďż˝Nďż˝uďż˝fďż˝$<ďż˝$%ďż˝Lďż˝Uďż˝Fďż˝$:ďż˝/4ďż˝6ďż˝6ďż˝6ďż˝6ďż˝$ďż˝aďż˝nďż˝_ďż˝%ďż˝ďż˝$?ďż˝&'ďż˝lďż˝]ďż˝ďż˝vďż˝$=ďż˝/4ďż˝6ďż˝6ďż˝6ďż˝6ďż˝ďż˝Qďż˝xďż˝xďż˝5ďż˝ďż˝8ďż˝8ďż˝+ďż˝+ďż˝ďż˝ďż˝8ďż˝8ďż˝qďż˝=ďż˝ r5cďż˝ďż˝ďż˝|jdkr|jdkr ||jzSt|tjďż˝ďż˝rt|j|jďż˝ďż˝|zS|jdkr ||jzS|t|ďż˝ďż˝zS)za ** br r) r rrrrrrr r!)r`rLs r4ďż˝__rpow__zFraction.__rpow__;sďż˝ďż˝ďż˝ďż˝>ďż˝Qďż˝ďż˝ďż˝1ďż˝<ďż˝1ďż˝#4ďż˝#4ďż˝ďż˝ďż˝ďż˝$ďż˝$ďż˝ďż˝aďż˝ďż˝)ďż˝*ďż˝*ďż˝ =ďż˝ďż˝Aďż˝Kďż˝ďż˝ďż˝7ďż˝7ďż˝1ďż˝<ďż˝<ďż˝ďż˝>ďż˝Qďż˝ďż˝ďż˝ďż˝ďż˝ďż˝$ďż˝$ďż˝ďż˝Eďż˝!ďż˝Hďż˝Hďż˝}ďż˝r5cďż˝:ďż˝t|j|jdďż˝ďż˝ďż˝S)z++a: Coerces a subclass instance to FractionFr ďż˝rrr rTs r4ďż˝__pos__zFraction.__pos__Isďż˝ďż˝ďż˝ďż˝ďż˝aďż˝nďż˝ďż˝Gďż˝Gďż˝Gďż˝Gr5cďż˝<ďż˝t|j|jdďż˝ďż˝ďż˝S)z-aFr rďż˝rTs r4ďż˝__neg__zFraction.__neg__Msďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝qďż˝~ďż˝%ďż˝Hďż˝Hďż˝Hďż˝Hr5cďż˝Tďż˝tt|jďż˝ďż˝|jdďż˝ďż˝ďż˝S)zabs(a)Fr )rrDrr rTs r4ďż˝__abs__zFraction.__abs__Qs#ďż˝ďż˝ďż˝ďż˝Aďż˝Lďż˝)ďż˝)ďż˝1ďż˝>ďż˝eďż˝Lďż˝Lďż˝Lďż˝Lr5cďż˝|ďż˝|jdkr||j|jzďż˝ďż˝S||j|jzďż˝ďż˝S)zint(a)rr)rLďż˝_indexs r4ďż˝__int__zFraction.__int__UsFďż˝ďż˝ďż˝<ďż˝!ďż˝ďż˝ďż˝ďż˝6ďż˝Qďż˝\ďż˝Mďż˝Qďż˝^ďż˝;ďż˝<ďż˝=ďż˝=ďż˝=ďż˝ďż˝6ďż˝!ďż˝,ďż˝!ďż˝.ďż˝8ďż˝9ďż˝9ďż˝9r5cďż˝Xďż˝|jdkr|j|jzS|j|jzS)z math.trunc(a)rrrTs r4ďż˝ __trunc__zFraction.__trunc__\s2ďż˝ďż˝ďż˝<ďż˝!ďż˝ďż˝ďż˝ďż˝lďż˝]ďż˝aďż˝nďż˝4ďż˝5ďż˝5ďż˝ďż˝<ďż˝1ďż˝>ďż˝1ďż˝1r5cďż˝ ďż˝|j|jzS)z math.floor(a)rďż˝rTs r4ďż˝ __floor__zFraction.__floor__csďż˝ďż˝ďż˝{ďż˝aďż˝mďż˝+ďż˝+r5cďż˝$ďż˝|j|jzS)zmath.ceil(a)rďż˝rTs r4ďż˝__ceil__zFraction.__ceil__gsďż˝ďż˝ďż˝+ďż˝ďż˝ďż˝ďż˝.ďż˝/ďż˝/r5cďż˝Zďż˝|ďż˝Pt|j|jďż˝ďż˝\}}|dz|jkr|S|dz|jkr|dzS|dzdkr|S|dzSdt|ďż˝ďż˝z}|dkr t t||zďż˝ďż˝|ďż˝ďż˝St t||zďż˝ďż˝|zďż˝ďż˝S)z?round(self, ndigits) Rounds half toward even. Nďż˝r rr)rďż˝rr rDrďż˝round)r/ďż˝ndigitsďż˝floorďż˝ remainderďż˝shifts r4ďż˝ __round__zFraction.__round__lsďż˝ďż˝ďż˝ ďż˝?ďż˝%ďż˝dďż˝nďż˝dďż˝6Fďż˝Gďż˝Gďż˝ďż˝Eďż˝9ďż˝ďż˝1ďż˝}ďż˝tďż˝/ďż˝/ďż˝/ďż˝ďż˝ďż˝ďż˝Qďż˝ďż˝ďż˝!1ďż˝1ďż˝1ďż˝ďż˝qďż˝yďż˝ ďż˝ďż˝ďż˝ďż˝aďż˝ďż˝ďż˝ďż˝ďż˝ďż˝qďż˝yďż˝ ďż˝ďż˝Cďż˝ďż˝Lďż˝Lďż˝ ďż˝ďż˝ďż˝Qďż˝;ďż˝;ďż˝ďż˝Eďż˝$ďż˝ďż˝,ďż˝/ďż˝/ďż˝ďż˝7ďż˝7ďż˝7ďż˝ďż˝Eďż˝$ďż˝ďż˝,ďż˝/ďż˝/ďż˝%ďż˝7ďż˝8ďż˝8ďż˝8r5cďż˝ďż˝ t|jdtďż˝ďż˝}ttt |jďż˝ďż˝ďż˝ďż˝|zďż˝ďż˝}n#t$r t}YnwxYw|jdkr|n|}|dkrdn|S)z hash(self)ďż˝ďż˝ďż˝ďż˝ďż˝rďż˝ďż˝ďż˝ďż˝ďż˝)ďż˝powr ďż˝_PyHASH_MODULUSďż˝hashrDrr&ďż˝_PyHASH_INF)r/ďż˝dinvďż˝hash_ďż˝results r4ďż˝__hash__zFraction.__hash__ďż˝sďż˝ďż˝ďż˝ <ďż˝ďż˝tďż˝(ďż˝"ďż˝oďż˝>ďż˝>ďż˝Dďż˝(ďż˝ďż˝cďż˝$ďż˝/ďż˝2ďż˝2ďż˝3ďż˝3ďż˝dďż˝:ďż˝;ďż˝;ďż˝Eďż˝Eďż˝ďż˝'ďż˝ ďż˝ ďż˝ ďż˝ďż˝Eďż˝Eďż˝Eďż˝ ďż˝ďż˝ďż˝ďż˝(ďż˝/ďż˝Qďż˝.ďż˝.ďż˝ďż˝ďż˝Uďż˝Fďż˝ďż˝ďż˝rďż˝\ďż˝\ďż˝rďż˝rďż˝vďż˝-sďż˝Aďż˝A#ďż˝"A#cďż˝ďż˝ďż˝t|ďż˝ďż˝tur|j|ko |jdkSt |t jďż˝ďż˝r |j|jko|j|jkSt |t j ďż˝ďż˝r|j dkr|j}t |tďż˝ďż˝rGtj|ďż˝ďż˝stj|ďż˝ďż˝rd|kS||ďż˝|ďż˝ďż˝kSt"S)za == br rďż˝)rrrr rrrrr rgďż˝imagďż˝realr!r,ďż˝isnanďż˝isinfr<r_rďż˝s r4ďż˝__eq__zFraction.__eq__ďż˝sďż˝ďż˝ďż˝ďż˝ďż˝7ďż˝7ďż˝cďż˝>ďż˝>ďż˝ďż˝<ďż˝1ďż˝$ďż˝<ďż˝ďż˝ďż˝1ďż˝)<ďż˝<ďż˝ďż˝aďż˝ďż˝)ďż˝*ďż˝*ďż˝ 5ďż˝ďż˝Lďż˝Aďż˝Kďż˝/ďż˝4ďż˝ďż˝Nďż˝aďż˝mďż˝3ďż˝ 5ďż˝ďż˝aďż˝ďż˝ďż˝)ďż˝)ďż˝ ďż˝aďż˝fďż˝ďż˝kďż˝kďż˝ďż˝ďż˝Aďż˝ďż˝aďż˝ďż˝ďż˝ďż˝ "ďż˝ďż˝zďż˝!ďż˝}ďż˝}ďż˝ ,ďż˝ďż˝ ďż˝1ďż˝ ďż˝ ďż˝ ,ďż˝ďż˝aďż˝xďż˝ďż˝ďż˝Aďż˝Lďż˝Lďż˝ďż˝Oďż˝Oďż˝+ďż˝+ďż˝"ďż˝!r5cďż˝`ďż˝t|tjďż˝ďż˝r&||j|jz|j|jzďż˝ďż˝St|tďż˝ďż˝rStj |ďż˝ďż˝stj |ďż˝ďż˝r|d|ďż˝ďż˝S|||ďż˝|ďż˝ďż˝ďż˝ďż˝StS)acHelper for comparison operators, for internal use only. Implement comparison between a Rational instance `self`, and either another Rational instance or a float `other`. If `other` is not a Rational instance or a float, return NotImplemented. `op` should be one of the six standard comparison operators. rďż˝) rrrrr r rr!r,rďż˝rďż˝r<r_)r/ďż˝otherďż˝ops r4ďż˝_richcmpzFraction._richcmpďż˝sďż˝ďż˝ďż˝ďż˝eďż˝Wďż˝-ďż˝.ďż˝.ďż˝ ;ďż˝ďż˝2ďż˝dďż˝oďż˝ďż˝(9ďż˝9ďż˝ďż˝'ďż˝%ďż˝/ďż˝9ďż˝;ďż˝;ďż˝ ;ďż˝ďż˝eďż˝Uďż˝#ďż˝#ďż˝ "ďż˝ďż˝zďż˝%ďż˝ ďż˝ ďż˝ 8ďż˝Dďż˝Jďż˝uďż˝$5ďż˝$5ďż˝ 8ďż˝ďż˝rďż˝#ďż˝uďż˝~ďż˝~ďż˝%ďż˝ďż˝rďż˝$ďż˝ďż˝ďż˝ďż˝ďż˝ 6ďż˝ 6ďż˝7ďż˝7ďż˝7ďż˝!ďż˝!r5cďż˝Bďż˝|ďż˝|tjďż˝ďż˝S)za < b)rďż˝ďż˝operatorďż˝ltrďż˝s r4ďż˝__lt__zFraction.__lt__ďż˝ďż˝ďż˝ďż˝ďż˝zďż˝zďż˝!ďż˝Xďż˝[ďż˝)ďż˝)ďż˝)r5cďż˝Bďż˝|ďż˝|tjďż˝ďż˝S)za > b)rďż˝rďż˝ďż˝gtrďż˝s r4ďż˝__gt__zFraction.__gt__ďż˝rďż˝r5cďż˝Bďż˝|ďż˝|tjďż˝ďż˝S)za <= b)rďż˝rďż˝ďż˝lerďż˝s r4ďż˝__le__zFraction.__le__ďż˝rďż˝r5cďż˝Bďż˝|ďż˝|tjďż˝ďż˝S)za >= b)rďż˝rďż˝ďż˝gerďż˝s r4ďż˝__ge__zFraction.__ge__ďż˝rďż˝r5cďż˝*ďż˝t|jďż˝ďż˝S)za != 0)ďż˝boolrrTs r4ďż˝__bool__zFraction.__bool__ďż˝sďż˝ďż˝ďż˝Aďż˝Lďż˝!ďż˝!ďż˝!r5cďż˝,ďż˝|j|j|jffSrS)r3rr rAs r4ďż˝ __reduce__zFraction.__reduce__ďż˝sďż˝ďż˝ďż˝ďż˝ďż˝ďż˝$ďż˝2Cďż˝ Dďż˝Eďż˝Er5cďż˝vďż˝t|ďż˝ďż˝tkr|S|ďż˝|j|jďż˝ďż˝SrSďż˝rrr3rr rAs r4ďż˝__copy__zFraction.__copy__ďż˝ďż˝1ďż˝ďż˝ďż˝ďż˝:ďż˝:ďż˝ďż˝!ďż˝!ďż˝ďż˝Kďż˝ďż˝~ďż˝~ďż˝dďż˝oďż˝tďż˝/@ďż˝Aďż˝Aďż˝Ar5cďż˝vďż˝t|ďż˝ďż˝tkr|S|ďż˝|j|jďż˝ďż˝SrSrďż˝)r/ďż˝memos r4ďż˝__deepcopy__zFraction.__deepcopy__ďż˝rďż˝r5)rN)rBrS)Er:ďż˝ __module__ďż˝__qualname__rjďż˝ __slots__rďż˝classmethodr<r?r"rQďż˝propertyrr rXr[rkrvrďż˝ďż˝addďż˝__add__ďż˝__radd__ryďż˝subďż˝__sub__ďż˝__rsub__r|ďż˝mulďż˝__mul__ďż˝__rmul__r~ďż˝truedivďż˝__truediv__ďż˝__rtruediv__rďż˝ďż˝floordivďż˝__floordiv__ďż˝ __rfloordiv__rďż˝rďż˝ďż˝ __divmod__ďż˝__rdivmod__rďż˝ďż˝modďż˝__mod__ďż˝__rmod__rďż˝rďż˝rďż˝rďż˝rďż˝ďż˝indexrďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝rďż˝ďż˝ __classcell__)r3s@r4rr&sďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝(/ďż˝Iďż˝hďż˝$ďż˝hďż˝hďż˝hďż˝hďż˝hďż˝hďż˝hďż˝Tďż˝*ďż˝*ďż˝ďż˝[ďż˝*ďż˝ďż˝ ,ďż˝ ,ďż˝ďż˝[ďż˝ ,ďż˝4ďż˝4ďż˝4ďż˝5ďż˝5ďż˝5ďż˝5ďż˝nďż˝ďż˝ďż˝ďż˝Xďż˝ďż˝ďż˝ďż˝ďż˝ďż˝Xďż˝ďż˝Cďż˝Cďż˝Cďż˝ Bďż˝Bďż˝Bďż˝i ďż˝i ďż˝i ďż˝^Cďż˝Cďż˝Cďż˝,ďż˝+ďż˝Dďż˝(ďż˝,ďż˝?ďż˝?ďż˝ďż˝Gďż˝Xďż˝Cďż˝Cďż˝Cďż˝,ďż˝+ďż˝Dďż˝(ďż˝,ďż˝?ďż˝?ďż˝ďż˝Gďż˝Xďż˝<ďż˝<ďż˝<ďż˝,ďż˝+ďż˝Dďż˝(ďż˝,ďż˝?ďż˝?ďż˝ďż˝Gďż˝Xďż˝0ďż˝0ďż˝0ďż˝$!4ďż˝ 3ďż˝Dďż˝(ďż˝:Jďż˝ Kďż˝ Kďż˝ďż˝Kďż˝ďż˝Nďż˝Nďż˝Nďż˝#6ďż˝"5ďż˝iďż˝ďż˝ARďż˝"Sďż˝"Sďż˝ďż˝Lďż˝-ďż˝-ďż˝-ďż˝-ďż˝2ďż˝1ďż˝'ďż˝6ďż˝Bďż˝Bďż˝ďż˝Jďż˝ďż˝Jďż˝Jďż˝Jďż˝ ,ďż˝+ďż˝Dďż˝(ďż˝,ďż˝?ďż˝?ďż˝ďż˝Gďż˝Xďż˝!ďż˝!ďż˝!ďż˝<ďż˝ďż˝ďż˝Hďż˝Hďż˝Hďż˝Iďż˝Iďż˝Iďż˝Mďż˝Mďż˝Mďż˝#ďż˝.ďż˝:ďż˝:ďż˝:ďż˝:ďż˝2ďż˝2ďż˝2ďż˝,ďż˝,ďż˝,ďż˝0ďż˝0ďż˝0ďż˝ 9ďż˝9ďż˝9ďż˝9ďż˝2.ďż˝.ďż˝.ďż˝B"ďż˝"ďż˝"ďż˝*"ďż˝"ďż˝"ďż˝,*ďż˝*ďż˝*ďż˝*ďż˝*ďż˝*ďż˝*ďż˝*ďż˝*ďż˝*ďż˝*ďż˝*ďż˝"ďż˝"ďż˝"ďż˝Fďż˝Fďż˝Fďż˝Bďż˝Bďż˝Bďż˝ Bďż˝Bďż˝Bďż˝Bďż˝Bďż˝Bďż˝Br5)rjrrr,rrďż˝ďż˝reďż˝sysďż˝__all__ďż˝ hash_infoďż˝modulusrďż˝ďż˝infrďż˝ďż˝compileďż˝VERBOSEďż˝ IGNORECASEr$rrďż˝r5r4ďż˝rďż˝sďż˝ďż˝ďż˝6ďż˝5ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ ďż˝ ďż˝ ďż˝ ďż˝ ďż˝ ďż˝ ďż˝ďż˝,ďż˝ďż˝ ďż˝-ďż˝'ďż˝ďż˝ďż˝mďż˝ďż˝ďż˝ďż˝2ďż˝:ďż˝ďż˝ďż˝Zďż˝"ďż˝-ďż˝ďż˝!ďż˝!ďż˝ďż˝NBďż˝NBďż˝NBďż˝NBďż˝NBďż˝wďż˝ďż˝NBďż˝NBďż˝NBďż˝NBďż˝NBr5