# Originally contributed by Sjoerd Mullender.
# Significantly modified by Jeffrey Yasskin <jyasskin at gmail.com>.
"""Fraction, infinite-precision, real numbers."""
from decimal import Decimal
import math
import numbers
import operator
import re
import sys
__all__ = ['Fraction', 'gcd']
def gcd(a, b):
"""Calculate the Greatest Common Divisor of a and b.
Unless b==0, the result will have the same sign as b (so that when
b is divided by it, the result comes out positive).
"""
import warnings
warnings.warn('fractions.gcd() is deprecated. Use math.gcd() instead.',
DeprecationWarning, 2)
if type(a) is int is type(b):
if (b or a) < 0:
return -math.gcd(a, b)
return math.gcd(a, b)
return _gcd(a, b)
def _gcd(a, b):
# Supports non-integers for backward compatibility.
while b:
a, b = b, a%b
return a
# Constants related to the hash implementation; hash(x) is based
# on the reduction of x modulo the prime _PyHASH_MODULUS.
_PyHASH_MODULUS = sys.hash_info.modulus
# Value to be used for rationals that reduce to infinity modulo
# _PyHASH_MODULUS.
_PyHASH_INF = sys.hash_info.inf
_RATIONAL_FORMAT = re.compile(r"""
\A\s* # optional whitespace at the start, then
(?P<sign>[-+]?) # an optional sign, then
(?=\d|\.\d) # lookahead for digit or .digit
(?P<num>\d*) # numerator (possibly empty)
(?: # followed by
(?:/(?P<denom>\d+))? # an optional denominator
| # or
(?:\.(?P<decimal>\d*))? # an optional fractional part
(?:E(?P<exp>[-+]?\d+))? # and optional exponent
)
\s*\Z # and optional whitespace to finish
""", re.VERBOSE | re.IGNORECASE)
class Fraction(numbers.Rational):
"""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)
"""
__slots__ = ('_numerator', '_denominator')
# We're immutable, so use __new__ not __init__
def __new__(cls, numerator=0, denominator=None, *, _normalize=True):
"""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)
"""
self = super(Fraction, cls).__new__(cls)
if denominator is None:
if type(numerator) is int:
self._numerator = numerator
self._denominator = 1
return self
elif isinstance(numerator, numbers.Rational):
self._numerator = numerator.numerator
self._denominator = numerator.denominator
return self
elif isinstance(numerator, (float, Decimal)):
# Exact conversion
self._numerator, self._denominator = numerator.as_integer_ratio()
return self
elif isinstance(numerator, str):
# Handle construction from strings.
m = _RATIONAL_FORMAT.match(numerator)
if m is None:
raise ValueError('Invalid literal for Fraction: %r' %
numerator)
numerator = int(m.group('num') or '0')
denom = m.group('denom')
if denom:
denominator = int(denom)
else:
denominator = 1
decimal = m.group('decimal')
if decimal:
scale = 10**len(decimal)
numerator = numerator * scale + int(decimal)
denominator *= scale
exp = m.group('exp')
if exp:
exp = int(exp)
if exp >= 0:
numerator *= 10**exp
else:
denominator *= 10**-exp
if m.group('sign') == '-':
numerator = -numerator
else:
raise TypeError("argument should be a string "
"or a Rational instance")
elif type(numerator) is int is type(denominator):
pass # *very* normal case
elif (isinstance(numerator, numbers.Rational) and
isinstance(denominator, numbers.Rational)):
numerator, denominator = (
numerator.numerator * denominator.denominator,
denominator.numerator * numerator.denominator
)
else:
raise TypeError("both arguments should be "
"Rational instances")
if denominator == 0:
raise ZeroDivisionError('Fraction(%s, 0)' % numerator)
if _normalize:
if type(numerator) is int is type(denominator):
# *very* normal case
g = math.gcd(numerator, denominator)
if denominator < 0:
g = -g
else:
g = _gcd(numerator, denominator)
numerator //= g
denominator //= g
self._numerator = numerator
self._denominator = denominator
return self
@classmethod
def from_float(cls, f):
"""Converts a finite float to a rational number, exactly.
Beware that Fraction.from_float(0.3) != Fraction(3, 10).
"""
if isinstance(f, numbers.Integral):
return cls(f)
elif not isinstance(f, float):
raise TypeError("%s.from_float() only takes floats, not %r (%s)" %
(cls.__name__, f, type(f).__name__))
return cls(*f.as_integer_ratio())
@classmethod
def from_decimal(cls, dec):
"""Converts a finite Decimal instance to a rational number, exactly."""
from decimal import Decimal
if isinstance(dec, numbers.Integral):
dec = Decimal(int(dec))
elif not isinstance(dec, Decimal):
raise TypeError(
"%s.from_decimal() only takes Decimals, not %r (%s)" %
(cls.__name__, dec, type(dec).__name__))
return cls(*dec.as_integer_ratio())
def limit_denominator(self, max_denominator=1000000):
"""Closest 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)
"""
# Algorithm notes: For any real number x, define a *best upper
# approximation* to x to be a rational number p/q such that:
#
# (1) p/q >= x, and
# (2) if p/q > r/s >= x then s > q, for any rational r/s.
#
# Define *best lower approximation* similarly. Then it can be
# proved that a rational number is a best upper or lower
# approximation to x if, and only if, it is a convergent or
# semiconvergent of the (unique shortest) continued fraction
# associated to x.
#
# To find a best rational approximation with denominator <= M,
# we find the best upper and lower approximations with
# denominator <= M and take whichever of these is closer to x.
# In the event of a tie, the bound with smaller denominator is
# chosen. If both denominators are equal (which can happen
# only when max_denominator == 1 and self is midway between
# two integers) the lower bound---i.e., the floor of self, is
# taken.
if max_denominator < 1:
raise ValueError("max_denominator should be at least 1")
if self._denominator <= max_denominator:
return Fraction(self)
p0, q0, p1, q1 = 0, 1, 1, 0
n, d = self._numerator, self._denominator
while True:
a = n//d
q2 = q0+a*q1
if q2 > max_denominator:
break
p0, q0, p1, q1 = p1, q1, p0+a*p1, q2
n, d = d, n-a*d
k = (max_denominator-q0)//q1
bound1 = Fraction(p0+k*p1, q0+k*q1)
bound2 = Fraction(p1, q1)
if abs(bound2 - self) <= abs(bound1-self):
return bound2
else:
return bound1
@property
def numerator(a):
return a._numerator
@property
def denominator(a):
return a._denominator
def __repr__(self):
"""repr(self)"""
return '%s(%s, %s)' % (self.__class__.__name__,
self._numerator, self._denominator)
def __str__(self):
"""str(self)"""
if self._denominator == 1:
return str(self._numerator)
else:
return '%s/%s' % (self._numerator, self._denominator)
def _operator_fallbacks(monomorphic_operator, fallback_operator):
"""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.
"""
def forward(a, b):
if isinstance(b, (int, Fraction)):
return monomorphic_operator(a, b)
elif isinstance(b, float):
return fallback_operator(float(a), b)
elif isinstance(b, complex):
return fallback_operator(complex(a), b)
else:
return NotImplemented
forward.__name__ = '__' + fallback_operator.__name__ + '__'
forward.__doc__ = monomorphic_operator.__doc__
def reverse(b, a):
if isinstance(a, numbers.Rational):
# Includes ints.
return monomorphic_operator(a, b)
elif isinstance(a, numbers.Real):
return fallback_operator(float(a), float(b))
elif isinstance(a, numbers.Complex):
return fallback_operator(complex(a), complex(b))
else:
return NotImplemented
reverse.__name__ = '__r' + fallback_operator.__name__ + '__'
reverse.__doc__ = monomorphic_operator.__doc__
return forward, reverse
def _add(a, b):
"""a + b"""
da, db = a.denominator, b.denominator
return Fraction(a.numerator * db + b.numerator * da,
da * db)
__add__, __radd__ = _operator_fallbacks(_add, operator.add)
def _sub(a, b):
"""a - b"""
da, db = a.denominator, b.denominator
return Fraction(a.numerator * db - b.numerator * da,
da * db)
__sub__, __rsub__ = _operator_fallbacks(_sub, operator.sub)
def _mul(a, b):
"""a * b"""
return Fraction(a.numerator * b.numerator, a.denominator * b.denominator)
__mul__, __rmul__ = _operator_fallbacks(_mul, operator.mul)
def _div(a, b):
"""a / b"""
return Fraction(a.numerator * b.denominator,
a.denominator * b.numerator)
__truediv__, __rtruediv__ = _operator_fallbacks(_div, operator.truediv)
def __floordiv__(a, b):
"""a // b"""
return math.floor(a / b)
def __rfloordiv__(b, a):
"""a // b"""
return math.floor(a / b)
def __mod__(a, b):
"""a % b"""
div = a // b
return a - b * div
def __rmod__(b, a):
"""a % b"""
div = a // b
return a - b * div
def __pow__(a, b):
"""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.
"""
if isinstance(b, numbers.Rational):
if b.denominator == 1:
power = b.numerator
if power >= 0:
return Fraction(a._numerator ** power,
a._denominator ** power,
_normalize=False)
elif a._numerator >= 0:
return Fraction(a._denominator ** -power,
a._numerator ** -power,
_normalize=False)
else:
return Fraction((-a._denominator) ** -power,
(-a._numerator) ** -power,
_normalize=False)
else:
# A fractional power will generally produce an
# irrational number.
return float(a) ** float(b)
else:
return float(a) ** b
def __rpow__(b, a):
"""a ** b"""
if b._denominator == 1 and b._numerator >= 0:
# If a is an int, keep it that way if possible.
return a ** b._numerator
if isinstance(a, numbers.Rational):
return Fraction(a.numerator, a.denominator) ** b
if b._denominator == 1:
return a ** b._numerator
return a ** float(b)
def __pos__(a):
"""+a: Coerces a subclass instance to Fraction"""
return Fraction(a._numerator, a._denominator, _normalize=False)
def __neg__(a):
"""-a"""
return Fraction(-a._numerator, a._denominator, _normalize=False)
def __abs__(a):
"""abs(a)"""
return Fraction(abs(a._numerator), a._denominator, _normalize=False)
def __trunc__(a):
"""trunc(a)"""
if a._numerator < 0:
return -(-a._numerator // a._denominator)
else:
return a._numerator // a._denominator
def __floor__(a):
"""Will be math.floor(a) in 3.0."""
return a.numerator // a.denominator
def __ceil__(a):
"""Will be math.ceil(a) in 3.0."""
# The negations cleverly convince floordiv to return the ceiling.
return -(-a.numerator // a.denominator)
def __round__(self, ndigits=None):
"""Will be round(self, ndigits) in 3.0.
Rounds half toward even.
"""
if ndigits is None:
floor, remainder = divmod(self.numerator, self.denominator)
if remainder * 2 < self.denominator:
return floor
elif remainder * 2 > self.denominator:
return floor + 1
# Deal with the half case:
elif floor % 2 == 0:
return floor
else:
return floor + 1
shift = 10**abs(ndigits)
# See _operator_fallbacks.forward to check that the results of
# these operations will always be Fraction and therefore have
# round().
if ndigits > 0:
return Fraction(round(self * shift), shift)
else:
return Fraction(round(self / shift) * shift)
def __hash__(self):
"""hash(self)"""
# XXX since this method is expensive, consider caching the result
# In order to make sure that the hash of a Fraction agrees
# with the hash of a numerically equal integer, float or
# Decimal instance, we follow the rules for numeric hashes
# outlined in the documentation. (See library docs, 'Built-in
# Types').
# dinv is the inverse of self._denominator modulo the prime
# _PyHASH_MODULUS, or 0 if self._denominator is divisible by
# _PyHASH_MODULUS.
dinv = pow(self._denominator, _PyHASH_MODULUS - 2, _PyHASH_MODULUS)
if not dinv:
hash_ = _PyHASH_INF
else:
hash_ = abs(self._numerator) * dinv % _PyHASH_MODULUS
result = hash_ if self >= 0 else -hash_
return -2 if result == -1 else result
def __eq__(a, b):
"""a == b"""
if type(b) is int:
return a._numerator == b and a._denominator == 1
if isinstance(b, numbers.Rational):
return (a._numerator == b.numerator and
a._denominator == b.denominator)
if isinstance(b, numbers.Complex) and b.imag == 0:
b = b.real
if isinstance(b, float):
if math.isnan(b) or math.isinf(b):
# comparisons with an infinity or nan should behave in
# the same way for any finite a, so treat a as zero.
return 0.0 == b
else:
return a == a.from_float(b)
else:
# Since a doesn't know how to compare with b, let's give b
# a chance to compare itself with a.
return NotImplemented
def _richcmp(self, other, op):
"""Helper 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.
"""
# convert other to a Rational instance where reasonable.
if isinstance(other, numbers.Rational):
return op(self._numerator * other.denominator,
self._denominator * other.numerator)
if isinstance(other, float):
if math.isnan(other) or math.isinf(other):
return op(0.0, other)
else:
return op(self, self.from_float(other))
else:
return NotImplemented
def __lt__(a, b):
"""a < b"""
return a._richcmp(b, operator.lt)
def __gt__(a, b):
"""a > b"""
return a._richcmp(b, operator.gt)
def __le__(a, b):
"""a <= b"""
return a._richcmp(b, operator.le)
def __ge__(a, b):
"""a >= b"""
return a._richcmp(b, operator.ge)
def __bool__(a):
"""a != 0"""
return a._numerator != 0
# support for pickling, copy, and deepcopy
def __reduce__(self):
return (self.__class__, (str(self),))
def __copy__(self):
if type(self) == Fraction:
return self # I'm immutable; therefore I am my own clone
return self.__class__(self._numerator, self._denominator)
def __deepcopy__(self, memo):
if type(self) == Fraction:
return self # My components are also immutable
return self.__class__(self._numerator, self._denominator)
I am a small town Minnesota single mom of two great kids who are my life. I began modeling 3 years ago for a photographer ho saw something in me I never did. Ice told me I should do one shoot with him and let that be the guide. I reluctantly agreed, and scheduled our date. I was sacred to death when he told me we would be doing a remake of the publicity stills of the 1956 movie Bus Stop, staring Marilyn Monroe. How in the world could I halfway resemble or pull off an icon the likes of Marylin Monroe in my first step in front of camera? Well, 2 hours later we had a nice set of images and I've been hooked ever since. We've done some really cool things and are looking hard at the future ahead to expand and get me out there a little more.
My pinup journey started at the age of 13 when I started collecting vintage decor and clothing- it has since spiraled into doing pinup shoots, meeting and developing friendships with other gorgeous pinups and being published in a pinup blog and magazine. Looking forward to the future and to see where other opportunities will take me!
Full Bio
I started getting into collecting vintage when I was a young kid, my mom would always take me into antique stores and this seemed to be what fueled it all. Eventually I started dressing and collecting vintage clothing and home decor. My apartment is now a great mix of MCM. I’ve done several pinup photoshoots and am looking to doing more in the future. I have been featured in a online pinup blog as well as being published in an state content creators magazine. Looking forward to the future and all the adventures it brings going forward.
Meet Belle Starr, your favorite tattooed 💉, curvy 💃 nurse turning heads and stealing hearts 💘 across Northwest Florida. A professional nurse 👩⚕️ during the week and a sultry pinup queen 👑 on the weekends, she’s the ultimate blend of classy ✨ and sassy 🔥—a vintage vixen with a modern twist.
Full Bio
Meet Belle Starr, your favorite tattooed 💉, curvy 💃 nurse turning heads and stealing hearts 💘 across Northwest Florida. A professional nurse 👩⚕️ during the week and a sultry pinup queen 👑 on the weekends, she’s the ultimate blend of classy ✨ and sassy 🔥—a vintage vixen with a modern twist.
She serves as the secretary for Pinups and Pumps Florida Chapter 💄 and is the official correspondent for PinupDatabase.com 🖋️. Belle Starr is dedicated to empowering women 👠, spotlighting the pinup community, and keeping the spirit of pinup history alive 📸. When she’s not hostessing 🎤 or interviewing at events 🌟, she’s a fierce advocate for the Ostel Place Foundation 🐴🐶🌿, a charity that helps people heal through horses, puppies, and the beauty of nature.
Whether she’s inspiring women 💋, enticing men 🕶️, or stealing the show as an event hostess 🎉, Belle Starr proves that beauty 💎, brains 🧠, and curves 🔥 never go out of style. Follow her journey for a dose of entertainment 🎭, empowerment 💪, and unforgettable vibes 🌟.
I'm a Pin Up model, classic car lover and Patriot. Been in Pin Up since 2014.
Full Bio
BoomBoom Bettie has been in the pinup world since 2014. She has participated in pageants in person and online since 2019. She loves the title of Favorite Pearl that she received. She is the founder of a Pin Up club called Black Sheep Pin Up Social Club in Arizona. She loves being a part of the pin up world and the sisterhood it creates. She loves to attend local car shows and Pin Up events.
𝑰 𝒑𝒐𝒔𝒕 my own pics, 𝒂𝒍𝒍 𝒄𝒍𝒂𝒔𝒔𝒚, 𝒖𝒔𝒖𝒂𝒍𝒍𝒚 𝒘𝒊𝒕𝒉 𝒇𝒖𝒏 𝒕𝒉𝒆𝒎𝒆𝒔. 𝑪𝒖𝒔𝒕𝒐𝒎 𝒘𝒐𝒓𝒌 𝒊𝒔 𝒂𝒗𝒂𝒊𝒍𝒂𝒃𝒍𝒆.
Jill of All, Owner of 5.
@currentteevents philanthropic tshirts
@shopcadesigns jewelry
@ciaraandruby dog models
@openmybar bar consulting
@calishamrock art/photography
My awesome journey began in California, followed by 25 wonderful years in Colorado. In 2019 I made the best choice of my life—moving to Florida, where I’ve truly found my home. The pin up community has been amazing, as I have always been drawn to the vibrant world of rockabilly style, classic cars, and music. Known for being kind, generous, and full of adventure, I cherish my experiences and connecting with new people. As a proud member of "Pinups and Pumps," I deeply appreciate the camaraderie with my sisters. Together, we give back through charity events, creating lasting bonds and memories.
Rating (average)
(0)
City
St. Augustine
Province
FL
Pin Up Group Membership
Pinups and Pumps Florida
Published in the Following Publications
Dream Beauty, Dream Pinup, Wonderland, Social Pin, Smitten Kitten, Dollface Digest, Crowns & Chrome, Drive In and many more
Clarice entered the pinup scene officially in 2019. Her first photoshoot was a tribute to the queen herself, Bettie Page. Dawning the same iconic bangs and hair darker than the devil's soul, she was a tattooed dead ringer. That photoshoot was featured in Retro Lovely's Bettie Page issue in 2019.
6 years later Clarice is a style of her own, finding herself more and more every day. She's a mental health advocate, constantly trying to educate about mental illness to help end the stigma. In March of this year she'll be celebrating 3 years free from alcohol. Supporting sobriety amongst her community is also a passion. Clarice is also Autistic, and tries to educate on hidden disabilities. Not only is she a pinup, she's a mommy first. Having 3 biological children, 3 "step"children, and her youngest being adopted, who's also autistic.
She enjoys creating art through painting, drawing, photography, and floral hair pieces.
Find her at the car shows, especially if there are rat rods and lowriders involved. Lowriders have been a part of her heart since high school. From being in a friend's hopper getting Taco Bell past her curfew, or cruising the beach with the systems bumping.
The name Clarice Von Darling is a tribute to The Silence of the Lambs. In her sister's memory.
62 year old trans woman who is now retired and living life to the fullest. Many past careers including dairy farmer firefighter/emt truck driver school bus driver church sexton cemetery sexton Public works director juice company truck driver and over the road truck driver. Two grown adult children ages 36 and 33 Two grand children ages 14 and 4 Local church member
