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2to3.html
58.11
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2019-03-27 12:19
__builtin__.html
10.81
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__future__.html
14.34
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__main__.html
7.34
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_winreg.html
63.12
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abc.html
25.22
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aepack.html
13.83
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aetools.html
16
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aetypes.html
20.53
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aifc.html
24.48
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al.html
18.23
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allos.html
34.47
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anydbm.html
17.7
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archiving.html
9.84
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argparse.html
257.44
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array.html
31.11
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ast.html
37.27
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asynchat.html
33.13
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asyncore.html
39.68
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atexit.html
18.02
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audioop.html
33.45
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autogil.html
8.53
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base64.html
21.35
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basehttpserver.html
36.56
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bastion.html
11.52
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bdb.html
40.42
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binascii.html
22.46
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binhex.html
11.04
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bisect.html
24.09
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bsddb.html
27.92
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bz2.html
28.37
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calendar.html
40.78
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carbon.html
50.78
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cd.html
29.33
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cgi.html
54.61
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cgihttpserver.html
13.7
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cgitb.html
11.98
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chunk.html
15.53
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cmath.html
27.7
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cmd.html
28.36
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code.html
26.33
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codecs.html
115.49
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codeop.html
15.53
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collections.html
144.19
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colorpicker.html
7.85
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colorsys.html
11.64
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commands.html
15.15
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compileall.html
18.18
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compiler.html
74.12
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configparser.html
66.16
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constants.html
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contextlib.html
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cookie.html
40.87
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cookielib.html
88.9
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copy.html
12.7
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copy_reg.html
14.35
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crypt.html
10.4
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crypto.html
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csv.html
74.66
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ctypes.html
258.37
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curses.ascii.html
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curses.html
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curses.panel.html
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custominterp.html
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datatypes.html
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datetime.html
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dbhash.html
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dbm.html
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debug.html
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decimal.html
217.1
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development.html
14.48
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difflib.html
89.82
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2019-03-27 12:19
dircache.html
12.01
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2019-03-27 12:19
dis.html
82.18
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distribution.html
7.5
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distutils.html
9.91
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dl.html
17.25
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doctest.html
181.53
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docxmlrpcserver.html
17.16
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2019-03-27 12:19
dumbdbm.html
15.18
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dummy_thread.html
9.83
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dummy_threading.html
8.72
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2019-03-27 12:19
easydialogs.html
32.35
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2019-03-27 12:19
email-examples.html
46.31
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2019-03-27 12:19
email.charset.html
28.73
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2019-03-27 12:19
email.encoders.html
12.55
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email.errors.html
17.53
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email.generator.html
22.6
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email.header.html
28.43
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email.html
54.42
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email.iterators.html
12.33
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email.message.html
68.54
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email.mime.html
30.91
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2019-03-27 12:19
email.parser.html
34.32
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2019-03-27 12:19
email.utils.html
26.44
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2019-03-27 12:19
ensurepip.html
17.9
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2019-03-27 12:19
errno.html
39.3
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2019-03-27 12:19
exceptions.html
62.2
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2019-03-27 12:19
fcntl.html
25.63
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filecmp.html
23.43
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2019-03-27 12:19
fileformats.html
9.43
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fileinput.html
26.44
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filesys.html
10.61
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fl.html
55.2
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2019-03-27 12:19
fm.html
12.7
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fnmatch.html
15.88
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2019-03-27 12:19
formatter.html
36.5
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fpectl.html
16.48
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fpformat.html
11.19
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2019-03-27 12:19
fractions.html
24.3
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framework.html
36.07
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frameworks.html
7.37
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ftplib.html
48.42
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functions.html
200.82
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functools.html
28.95
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future_builtins.html
14.14
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gc.html
27.76
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gdbm.html
17.39
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gensuitemodule.html
12.28
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getopt.html
24.69
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getpass.html
11.16
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gettext.html
82.98
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gl.html
23.77
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glob.html
14.09
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grp.html
11.05
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gzip.html
20.09
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hashlib.html
24.87
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2019-03-27 12:19
heapq.html
34.07
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hmac.html
14.04
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hotshot.html
19.67
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htmllib.html
27.03
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htmlparser.html
41.44
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httplib.html
69.27
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i18n.html
9.81
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ic.html
18.22
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idle.html
41.16
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imageop.html
15.72
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imaplib.html
57.14
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imgfile.html
12.43
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imghdr.html
11.95
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imp.html
36.72
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importlib.html
8.72
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imputil.html
32.78
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index.html
77.23
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inspect.html
55.49
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internet.html
25.53
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intro.html
9.13
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io.html
111.04
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ipc.html
16.21
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itertools.html
122.46
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jpeg.html
13.43
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json.html
71.95
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keyword.html
8.02
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language.html
11.41
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linecache.html
11.15
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locale.html
60.13
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logging.config.html
78.17
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logging.handlers.html
78.31
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logging.html
107.67
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mac.html
22.83
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macos.html
15.75
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macosa.html
13.75
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macostools.html
16.5
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macpath.html
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mailbox.html
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mailcap.html
13.8
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markup.html
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marshal.html
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math.html
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md5.html
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mhlib.html
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mimetools.html
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mimetypes.html
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mimewriter.html
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mimify.html
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miniaeframe.html
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misc.html
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mm.html
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mmap.html
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modulefinder.html
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modules.html
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msilib.html
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msvcrt.html
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multifile.html
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multiprocessing.html
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mutex.html
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netdata.html
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netrc.html
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new.html
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nis.html
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nntplib.html
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numbers.html
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numeric.html
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operator.html
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optparse.html
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os.html
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os.path.html
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ossaudiodev.html
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othergui.html
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parser.html
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pdb.html
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persistence.html
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pickle.html
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pickletools.html
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pipes.html
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pkgutil.html
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platform.html
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plistlib.html
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popen2.html
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poplib.html
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posix.html
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posixfile.html
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pprint.html
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profile.html
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pty.html
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pwd.html
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py_compile.html
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pyclbr.html
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pydoc.html
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pyexpat.html
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python.html
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queue.html
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quopri.html
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random.html
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re.html
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readline.html
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repr.html
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resource.html
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restricted.html
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rexec.html
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rfc822.html
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rlcompleter.html
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robotparser.html
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runpy.html
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sched.html
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scrolledtext.html
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select.html
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sets.html
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sgi.html
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sgmllib.html
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sha.html
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shelve.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="X-UA-Compatible" content="IE=Edge" /> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>9.5. fractions — Rational numbers — Python 2.7.16 documentation</title> <link rel="stylesheet" href="../_static/classic.css" type="text/css" /> <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> <script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script> <script type="text/javascript" src="../_static/jquery.js"></script> <script type="text/javascript" src="../_static/underscore.js"></script> <script type="text/javascript" src="../_static/doctools.js"></script> <script type="text/javascript" src="../_static/sidebar.js"></script> <link rel="search" type="application/opensearchdescription+xml" title="Search within Python 2.7.16 documentation" href="../_static/opensearch.xml"/> <link rel="author" title="About these documents" href="../about.html" /> <link rel="index" title="Index" href="../genindex.html" /> <link rel="search" title="Search" href="../search.html" /> <link rel="copyright" title="Copyright" href="../copyright.html" /> <link rel="next" title="9.6. random — Generate pseudo-random numbers" href="random.html" /> <link rel="prev" title="9.4. decimal — Decimal fixed point and floating point arithmetic" href="decimal.html" /> <link rel="shortcut icon" type="image/png" href="../_static/py.png" /> <link rel="canonical" href="https://docs.python.org/2/library/fractions.html" /> <script type="text/javascript" src="../_static/copybutton.js"></script> </head><body> <div class="related" role="navigation" aria-label="related navigation"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" accesskey="I">index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="random.html" title="9.6. random — Generate pseudo-random numbers" accesskey="N">next</a> |</li> <li class="right" > <a href="decimal.html" title="9.4. decimal — Decimal fixed point and floating point arithmetic" accesskey="P">previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="https://www.python.org/">Python</a> »</li> <li> <a href="../index.html">Python 2.7.16 documentation</a> » </li> <li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> »</li> <li class="nav-item nav-item-2"><a href="numeric.html" accesskey="U">9. Numeric and Mathematical Modules</a> »</li> </ul> </div> <div class="document"> <div class="documentwrapper"> <div class="bodywrapper"> <div class="body" role="main"> <div class="section" id="module-fractions"> <span id="fractions-rational-numbers"></span><h1>9.5. <a class="reference internal" href="#module-fractions" title="fractions: Rational numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">fractions</span></code></a> — Rational numbers<a class="headerlink" href="#module-fractions" title="Permalink to this headline">¶</a></h1> <div class="versionadded"> <p><span class="versionmodified">New in version 2.6.</span></p> </div> <p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/2.7/Lib/fractions.py">Lib/fractions.py</a></p> <hr class="docutils" /> <p>The <a class="reference internal" href="#module-fractions" title="fractions: Rational numbers."><code class="xref py py-mod docutils literal notranslate"><span class="pre">fractions</span></code></a> module provides support for rational number arithmetic.</p> <p>A Fraction instance can be constructed from a pair of integers, from another rational number, or from a string.</p> <dl class="class"> <dt id="fractions.Fraction"> <em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>numerator=0</em>, <em>denominator=1</em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction" title="Permalink to this definition">¶</a></dt> <dt> <em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>other_fraction</em><span class="sig-paren">)</span></dt> <dt> <em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>float</em><span class="sig-paren">)</span></dt> <dt> <em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>decimal</em><span class="sig-paren">)</span></dt> <dt> <em class="property">class </em><code class="descclassname">fractions.</code><code class="descname">Fraction</code><span class="sig-paren">(</span><em>string</em><span class="sig-paren">)</span></dt> <dd><p>The first version requires that <em>numerator</em> and <em>denominator</em> are instances of <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Rational</span></code></a> and returns a new <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance with value <code class="docutils literal notranslate"><span class="pre">numerator/denominator</span></code>. If <em>denominator</em> is <code class="xref py py-const docutils literal notranslate"><span class="pre">0</span></code>, it raises a <a class="reference internal" href="exceptions.html#exceptions.ZeroDivisionError" title="exceptions.ZeroDivisionError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ZeroDivisionError</span></code></a>. The second version requires that <em>other_fraction</em> is an instance of <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Rational</span></code></a> and returns a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance with the same value. The next two versions accept either a <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> or a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> instance, and return a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance with exactly the same value. Note that due to the usual issues with binary floating-point (see <a class="reference internal" href="../tutorial/floatingpoint.html#tut-fp-issues"><span class="std std-ref">Floating Point Arithmetic: Issues and Limitations</span></a>), the argument to <code class="docutils literal notranslate"><span class="pre">Fraction(1.1)</span></code> is not exactly equal to 11/10, and so <code class="docutils literal notranslate"><span class="pre">Fraction(1.1)</span></code> does <em>not</em> return <code class="docutils literal notranslate"><span class="pre">Fraction(11,</span> <span class="pre">10)</span></code> as one might expect. (But see the documentation for the <a class="reference internal" href="#fractions.Fraction.limit_denominator" title="fractions.Fraction.limit_denominator"><code class="xref py py-meth docutils literal notranslate"><span class="pre">limit_denominator()</span></code></a> method below.) The last version of the constructor expects a string or unicode instance. The usual form for this instance is:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="p">[</span><span class="n">sign</span><span class="p">]</span> <span class="n">numerator</span> <span class="p">[</span><span class="s1">'/'</span> <span class="n">denominator</span><span class="p">]</span> </pre></div> </div> <p>where the optional <code class="docutils literal notranslate"><span class="pre">sign</span></code> may be either ‘+’ or ‘-‘ and <code class="docutils literal notranslate"><span class="pre">numerator</span></code> and <code class="docutils literal notranslate"><span class="pre">denominator</span></code> (if present) are strings of decimal digits. In addition, any string that represents a finite value and is accepted by the <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> constructor is also accepted by the <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> constructor. In either form the input string may also have leading and/or trailing whitespace. Here are some examples:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="k">import</span> <span class="n">Fraction</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mi">16</span><span class="p">,</span> <span class="o">-</span><span class="mi">10</span><span class="p">)</span> <span class="go">Fraction(-8, 5)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mi">123</span><span class="p">)</span> <span class="go">Fraction(123, 1)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">()</span> <span class="go">Fraction(0, 1)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">'3/7'</span><span class="p">)</span> <span class="go">Fraction(3, 7)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">' -3/7 '</span><span class="p">)</span> <span class="go">Fraction(-3, 7)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">'1.414213 </span><span class="se">\t\n</span><span class="s1">'</span><span class="p">)</span> <span class="go">Fraction(1414213, 1000000)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">'-.125'</span><span class="p">)</span> <span class="go">Fraction(-1, 8)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">'7e-6'</span><span class="p">)</span> <span class="go">Fraction(7, 1000000)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">2.25</span><span class="p">)</span> <span class="go">Fraction(9, 4)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">1.1</span><span class="p">)</span> <span class="go">Fraction(2476979795053773, 2251799813685248)</span> <span class="gp">>>> </span><span class="kn">from</span> <span class="nn">decimal</span> <span class="k">import</span> <span class="n">Decimal</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="n">Decimal</span><span class="p">(</span><span class="s1">'1.1'</span><span class="p">))</span> <span class="go">Fraction(11, 10)</span> </pre></div> </div> <p>The <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> class inherits from the abstract base class <a class="reference internal" href="numbers.html#numbers.Rational" title="numbers.Rational"><code class="xref py py-class docutils literal notranslate"><span class="pre">numbers.Rational</span></code></a>, and implements all of the methods and operations from that class. <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instances are hashable, and should be treated as immutable. In addition, <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> has the following methods:</p> <div class="versionchanged"> <p><span class="versionmodified">Changed in version 2.7: </span>The <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> constructor now accepts <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> and <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> instances.</p> </div> <dl class="method"> <dt id="fractions.Fraction.from_float"> <code class="descname">from_float</code><span class="sig-paren">(</span><em>flt</em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.from_float" title="Permalink to this definition">¶</a></dt> <dd><p>This class method constructs a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> representing the exact value of <em>flt</em>, which must be a <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>. Beware that <code class="docutils literal notranslate"><span class="pre">Fraction.from_float(0.3)</span></code> is not the same value as <code class="docutils literal notranslate"><span class="pre">Fraction(3,</span> <span class="pre">10)</span></code>.</p> <div class="admonition note"> <p class="first admonition-title">Note</p> <p class="last">From Python 2.7 onwards, you can also construct a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance directly from a <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>.</p> </div> </dd></dl> <dl class="method"> <dt id="fractions.Fraction.from_decimal"> <code class="descname">from_decimal</code><span class="sig-paren">(</span><em>dec</em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.from_decimal" title="Permalink to this definition">¶</a></dt> <dd><p>This class method constructs a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> representing the exact value of <em>dec</em>, which must be a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a>.</p> <div class="admonition note"> <p class="first admonition-title">Note</p> <p class="last">From Python 2.7 onwards, you can also construct a <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> instance directly from a <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate"><span class="pre">decimal.Decimal</span></code></a> instance.</p> </div> </dd></dl> <dl class="method"> <dt id="fractions.Fraction.limit_denominator"> <code class="descname">limit_denominator</code><span class="sig-paren">(</span><em>max_denominator=1000000</em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.Fraction.limit_denominator" title="Permalink to this definition">¶</a></dt> <dd><p>Finds and returns the closest <a class="reference internal" href="#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate"><span class="pre">Fraction</span></code></a> to <code class="docutils literal notranslate"><span class="pre">self</span></code> that has denominator at most max_denominator. This method is useful for finding rational approximations to a given floating-point number:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">fractions</span> <span class="k">import</span> <span class="n">Fraction</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="s1">'3.1415926535897932'</span><span class="p">)</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">(</span><span class="mi">1000</span><span class="p">)</span> <span class="go">Fraction(355, 113)</span> </pre></div> </div> <p>or for recovering a rational number that’s represented as a float:</p> <div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">math</span> <span class="k">import</span> <span class="n">pi</span><span class="p">,</span> <span class="n">cos</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span> <span class="go">Fraction(4503599627370497, 9007199254740992)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="n">cos</span><span class="p">(</span><span class="n">pi</span><span class="o">/</span><span class="mi">3</span><span class="p">))</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">()</span> <span class="go">Fraction(1, 2)</span> <span class="gp">>>> </span><span class="n">Fraction</span><span class="p">(</span><span class="mf">1.1</span><span class="p">)</span><span class="o">.</span><span class="n">limit_denominator</span><span class="p">()</span> <span class="go">Fraction(11, 10)</span> </pre></div> </div> </dd></dl> </dd></dl> <dl class="function"> <dt id="fractions.gcd"> <code class="descclassname">fractions.</code><code class="descname">gcd</code><span class="sig-paren">(</span><em>a</em>, <em>b</em><span class="sig-paren">)</span><a class="headerlink" href="#fractions.gcd" title="Permalink to this definition">¶</a></dt> <dd><p>Return the greatest common divisor of the integers <em>a</em> and <em>b</em>. If either <em>a</em> or <em>b</em> is nonzero, then the absolute value of <code class="docutils literal notranslate"><span class="pre">gcd(a,</span> <span class="pre">b)</span></code> is the largest integer that divides both <em>a</em> and <em>b</em>. <code class="docutils literal notranslate"><span class="pre">gcd(a,b)</span></code> has the same sign as <em>b</em> if <em>b</em> is nonzero; otherwise it takes the sign of <em>a</em>. <code class="docutils literal notranslate"><span class="pre">gcd(0,</span> <span class="pre">0)</span></code> returns <code class="docutils literal notranslate"><span class="pre">0</span></code>.</p> </dd></dl> <div class="admonition seealso"> <p class="first admonition-title">See also</p> <dl class="last docutils"> <dt>Module <a class="reference internal" href="numbers.html#module-numbers" title="numbers: Numeric abstract base classes (Complex, Real, Integral, etc.)."><code class="xref py py-mod docutils literal notranslate"><span class="pre">numbers</span></code></a></dt> <dd>The abstract base classes making up the numeric tower.</dd> </dl> </div> </div> </div> </div> </div> <div class="sphinxsidebar" role="navigation" aria-label="main navigation"> <div class="sphinxsidebarwrapper"> <h4>Previous topic</h4> <p class="topless"><a href="decimal.html" title="previous chapter">9.4. <code class="docutils literal notranslate"><span class="pre">decimal</span></code> — Decimal fixed point and floating point arithmetic</a></p> <h4>Next topic</h4> <p class="topless"><a href="random.html" title="next chapter">9.6. <code class="docutils literal notranslate"><span class="pre">random</span></code> — Generate pseudo-random numbers</a></p> <div role="note" aria-label="source link"> <h3>This Page</h3> <ul class="this-page-menu"> <li><a href="../_sources/library/fractions.rst.txt" rel="nofollow">Show Source</a></li> </ul> </div> <div id="searchbox" style="display: none" role="search"> <h3>Quick search</h3> <div class="searchformwrapper"> <form class="search" action="../search.html" method="get"> <input type="text" name="q" /> <input type="submit" value="Go" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form> </div> </div> <script type="text/javascript">$('#searchbox').show(0);</script> </div> </div> <div class="clearer"></div> </div> <div class="related" role="navigation" aria-label="related navigation"> <h3>Navigation</h3> <ul> <li class="right" style="margin-right: 10px"> <a href="../genindex.html" title="General Index" >index</a></li> <li class="right" > <a href="../py-modindex.html" title="Python Module Index" >modules</a> |</li> <li class="right" > <a href="random.html" title="9.6. random — Generate pseudo-random numbers" >next</a> |</li> <li class="right" > <a href="decimal.html" title="9.4. decimal — Decimal fixed point and floating point arithmetic" >previous</a> |</li> <li><img src="../_static/py.png" alt="" style="vertical-align: middle; margin-top: -1px"/></li> <li><a href="https://www.python.org/">Python</a> »</li> <li> <a href="../index.html">Python 2.7.16 documentation</a> » </li> <li class="nav-item nav-item-1"><a href="index.html" >The Python Standard Library</a> »</li> <li class="nav-item nav-item-2"><a href="numeric.html" >9. 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