1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
|
:mod:`fractions` --- Rational numbers
=====================================
.. module:: fractions
:synopsis: Rational numbers.
.. moduleauthor:: Jeffrey Yasskin <jyasskin at gmail.com>
.. sectionauthor:: Jeffrey Yasskin <jyasskin at gmail.com>
**Source code:** :source:`Lib/fractions.py`
--------------
The :mod:`fractions` module provides support for rational number arithmetic.
A Fraction instance can be constructed from a pair of integers, from
another rational number, or from a string.
.. class:: Fraction(numerator=0, denominator=1)
Fraction(other_fraction)
Fraction(float)
Fraction(decimal)
Fraction(string)
The first version requires that *numerator* and *denominator* are instances
of :class:`numbers.Rational` and returns a new :class:`Fraction` instance
with value ``numerator/denominator``. If *denominator* is :const:`0`, it
raises a :exc:`ZeroDivisionError`. The second version requires that
*other_fraction* is an instance of :class:`numbers.Rational` and returns a
:class:`Fraction` instance with the same value. The next two versions accept
either a :class:`float` or a :class:`decimal.Decimal` instance, and return a
:class:`Fraction` instance with exactly the same value. Note that due to the
usual issues with binary floating-point (see :ref:`tut-fp-issues`), the
argument to ``Fraction(1.1)`` is not exactly equal to 11/10, and so
``Fraction(1.1)`` does *not* return ``Fraction(11, 10)`` as one might expect.
(But see the documentation for the :meth:`limit_denominator` method below.)
The last version of the constructor expects a string or unicode instance.
The usual form for this instance is::
[sign] numerator ['/' denominator]
where the optional ``sign`` may be either '+' or '-' and
``numerator`` and ``denominator`` (if present) are strings of
decimal digits (underscores may be used to delimit digits as with
integral literals in code). In addition, any string that represents a finite
value and is accepted by the :class:`float` constructor is also
accepted by the :class:`Fraction` constructor. In either form the
input string may also have leading and/or trailing whitespace.
Here are some examples::
>>> from fractions import Fraction
>>> Fraction(16, -10)
Fraction(-8, 5)
>>> Fraction(123)
Fraction(123, 1)
>>> Fraction()
Fraction(0, 1)
>>> Fraction('3/7')
Fraction(3, 7)
>>> Fraction(' -3/7 ')
Fraction(-3, 7)
>>> Fraction('1.414213 \t\n')
Fraction(1414213, 1000000)
>>> Fraction('-.125')
Fraction(-1, 8)
>>> Fraction('7e-6')
Fraction(7, 1000000)
>>> Fraction(2.25)
Fraction(9, 4)
>>> Fraction(1.1)
Fraction(2476979795053773, 2251799813685248)
>>> from decimal import Decimal
>>> Fraction(Decimal('1.1'))
Fraction(11, 10)
The :class:`Fraction` class inherits from the abstract base class
:class:`numbers.Rational`, and implements all of the methods and
operations from that class. :class:`Fraction` instances are hashable,
and should be treated as immutable. In addition,
:class:`Fraction` has the following properties and methods:
.. versionchanged:: 3.2
The :class:`Fraction` constructor now accepts :class:`float` and
:class:`decimal.Decimal` instances.
.. versionchanged:: 3.9
The :func:`math.gcd` function is now used to normalize the *numerator*
and *denominator*. :func:`math.gcd` always return a :class:`int` type.
Previously, the GCD type depended on *numerator* and *denominator*.
.. versionchanged:: 3.11
Underscores are now permitted when creating a :class:`Fraction` instance
from a string, following :PEP:`515` rules.
.. versionchanged:: 3.11
:class:`Fraction` implements ``__int__`` now to satisfy
``typing.SupportsInt`` instance checks.
.. versionchanged:: 3.12
Space is allowed around the slash for string inputs: ``Fraction('2 / 3')``.
.. attribute:: numerator
Numerator of the Fraction in lowest term.
.. attribute:: denominator
Denominator of the Fraction in lowest term.
.. method:: as_integer_ratio()
Return a tuple of two integers, whose ratio is equal
to the Fraction and with a positive denominator.
.. versionadded:: 3.8
.. method:: is_integer()
Return ``True`` if the Fraction is an integer.
.. versionadded:: 3.12
.. classmethod:: from_float(flt)
Alternative constructor which only accepts instances of
:class:`float` or :class:`numbers.Integral`. Beware that
``Fraction.from_float(0.3)`` is not the same value as ``Fraction(3, 10)``.
.. note::
From Python 3.2 onwards, you can also construct a
:class:`Fraction` instance directly from a :class:`float`.
.. classmethod:: from_decimal(dec)
Alternative constructor which only accepts instances of
:class:`decimal.Decimal` or :class:`numbers.Integral`.
.. note::
From Python 3.2 onwards, you can also construct a
:class:`Fraction` instance directly from a :class:`decimal.Decimal`
instance.
.. method:: limit_denominator(max_denominator=1000000)
Finds and returns the closest :class:`Fraction` to ``self`` that has
denominator at most max_denominator. This method is useful for finding
rational approximations to a given floating-point number:
>>> from fractions import Fraction
>>> Fraction('3.1415926535897932').limit_denominator(1000)
Fraction(355, 113)
or for recovering a rational number that's represented as a float:
>>> from math import pi, cos
>>> Fraction(cos(pi/3))
Fraction(4503599627370497, 9007199254740992)
>>> Fraction(cos(pi/3)).limit_denominator()
Fraction(1, 2)
>>> Fraction(1.1).limit_denominator()
Fraction(11, 10)
.. method:: __floor__()
Returns the greatest :class:`int` ``<= self``. This method can
also be accessed through the :func:`math.floor` function:
>>> from math import floor
>>> floor(Fraction(355, 113))
3
.. method:: __ceil__()
Returns the least :class:`int` ``>= self``. This method can
also be accessed through the :func:`math.ceil` function.
.. method:: __round__()
__round__(ndigits)
The first version returns the nearest :class:`int` to ``self``,
rounding half to even. The second version rounds ``self`` to the
nearest multiple of ``Fraction(1, 10**ndigits)`` (logically, if
``ndigits`` is negative), again rounding half toward even. This
method can also be accessed through the :func:`round` function.
.. seealso::
Module :mod:`numbers`
The abstract base classes making up the numeric tower.
|
