fractions.py: Rational Number Arithmetic

fractions.py implements exact rational arithmetic. Every Fraction is stored as a normalized numerator and denominator pair of arbitrary-precision integers. The file has no C accelerator; this single source is the whole implementation.

Map

Lines	Symbol	Purpose
1–40	imports, __all__	math, numbers, operator, re
41–90	_RATIONAL_FORMAT	Regex for parsing fraction strings like "3/4" or "-1.5"
91–200	Fraction.__new__	Overloaded constructor for int, float, Decimal, str, Rational
201–260	_gcd, normalization	Sign canonicalization, math.gcd call, zero handling
261–360	from_float, from_decimal	Exact conversion via float.as_integer_ratio
361–450	limit_denominator	Stern-Brocot / best-rational-approximation loop
451–540	__add__, __sub__, __mul__, __truediv__	Cross-multiply with math.gcd reduction via lcm
541–600	__floordiv__, __mod__, __divmod__	Delegate to integer floor after exact division
601–650	__floor__, __ceil__, __round__	Use // and tie-breaking for __round__
651–700	Comparisons, __hash__, __repr__	Interop with float and Decimal comparisons

Reading

Constructor overloads and normalization

Fraction.__new__ accepts six distinct input shapes. For a (numerator, denominator) pair it calls math.gcd and divides both values, then moves the sign to the numerator. For float input it delegates to float.as_integer_ratio(), which returns an exact (int, int) pair without any floating-point rounding. For str input it applies _RATIONAL_FORMAT and parses the matched groups.

Arithmetic via lcm

__add__ and __sub__ compute a common denominator using lcm = a_denom // math.gcd(a_denom, b_denom) * b_denom to minimize intermediate integer size before calling Fraction(numerator, lcm). __mul__ does a cross-GCD reduction before multiplying to keep coefficients small throughout a chain of operations. These patterns avoid catastrophic coefficient growth in long accumulations.

limit_denominator

The algorithm walks the Stern-Brocot tree, maintaining a mediant bracket (p0/q0, p1/q1) and updating bounds by comparing the target value against the mediant. It terminates when q1 > max_denominator, then picks whichever endpoint is closer to the original value. This gives the best rational approximation for any given denominator bound.

gopy notes

• Fraction depends on math.gcd (ported in module/math) and arbitrary-precision integers. In gopy, big integers go through objects/int.go which wraps math/big.
• __hash__ must satisfy hash(Fraction(n, 1)) == hash(n) and hash(Fraction(x)) == hash(x) for any float x that is an exact integer. The hash formula from numbers.Rational must be reproduced exactly.
• 3.14 deprecated passing a float or Decimal as the sole positional argument without a keyword; Fraction(0.5) now emits DeprecationWarning. The port must replicate the warning path.
• _RATIONAL_FORMAT uses named groups ((?P<sign>...)) that map directly to Go's regexp named submatches via SubexpIndex.