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# SOME DESCRIPTIVE TITLE.
# Copyright (C) 2001-2018, Python Software Foundation
# This file is distributed under the same license as the Python package.
#
# Translators:
msgid ""
msgstr ""
"Project-Id-Version: Python 3.7\n"
"Report-Msgid-Bugs-To: \n"
"POT-Creation-Date: 2018-06-26 18:54+0800\n"
"PO-Revision-Date: 2018-11-20 14:26+0800\n"
"Last-Translator: Adrian Liaw <adrianliaw2000@gmail.com>\n"
"Language-Team: Chinese - TAIWAN (https://github.com/python/python-docs-zh-"
"tw)\n"
"Language: zh_TW\n"
"MIME-Version: 1.0\n"
"Content-Type: text/plain; charset=UTF-8\n"
"Content-Transfer-Encoding: 8bit\n"
"Plural-Forms: nplurals=1; plural=0;\n"
"X-Generator: Poedit 2.1.1\n"
#: ../../tutorial/floatingpoint.rst:9
msgid "Floating Point Arithmetic: Issues and Limitations"
msgstr "浮點數運算：問題與限制"
#: ../../tutorial/floatingpoint.rst:14
msgid ""
"Floating-point numbers are represented in computer hardware as base 2 "
"(binary) fractions. For example, the decimal fraction ::"
msgstr ""
"在計算機架構中，浮點數透過二進位小數表示。例如說，在十進位小數中：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:19
msgid ""
"has value 1/10 + 2/100 + 5/1000, and in the same way the binary fraction ::"
msgstr ""
"可被分為 1/10 + 2/100 + 5/1000，同樣的道理，二進位小數 ：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:23
msgid ""
"has value 0/2 + 0/4 + 1/8. These two fractions have identical values, the "
"only real difference being that the first is written in base 10 fractional "
"notation, and the second in base 2."
msgstr ""
"可被分為 0/2 + 0/4 + 1/8。這兩個小數有相同的數值，而唯一真正的不同在於前者以"
"十進位表示，後者以二進位表示。"
#: ../../tutorial/floatingpoint.rst:27
msgid ""
"Unfortunately, most decimal fractions cannot be represented exactly as "
"binary fractions. A consequence is that, in general, the decimal floating-"
"point numbers you enter are only approximated by the binary floating-point "
"numbers actually stored in the machine."
msgstr ""
"不幸的是，大多數十進位小數無法精準地以二進位小數表示。一般的結果為，您輸入的"
"十進位浮點數由實際存在計算機中的二進位浮點數近似。"
#: ../../tutorial/floatingpoint.rst:32
msgid ""
"The problem is easier to understand at first in base 10. Consider the "
"fraction 1/3. You can approximate that as a base 10 fraction::"
msgstr ""
"在十進位中，這個問題更容易被理解。以分數 1/3 為例，您可以將其近似為十進位小"
"數：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:37 ../../tutorial/floatingpoint.rst:41
msgid "or, better, ::"
msgstr ""
"或者，更好的近似：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:45
msgid ""
"and so on. No matter how many digits you're willing to write down, the "
"result will never be exactly 1/3, but will be an increasingly better "
"approximation of 1/3."
msgstr ""
"依此類推，不論你使用多少位數表示小數，最後的結果都無法精準地表示 1/3，但你還"
"是能越來越精準地表示 1/3。"
#: ../../tutorial/floatingpoint.rst:49
msgid ""
"In the same way, no matter how many base 2 digits you're willing to use, the "
"decimal value 0.1 cannot be represented exactly as a base 2 fraction. In "
"base 2, 1/10 is the infinitely repeating fraction ::"
msgstr ""
"同樣的道理，不論你願意以多少位數表示二進位小數，十進位小數 0.1 都無法被二進位"
"小數精準地表達。在二進位小數中， 1/10 會是一個無限循環小數：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:55
msgid ""
"Stop at any finite number of bits, and you get an approximation. On most "
"machines today, floats are approximated using a binary fraction with the "
"numerator using the first 53 bits starting with the most significant bit and "
"with the denominator as a power of two. In the case of 1/10, the binary "
"fraction is ``3602879701896397 / 2 ** 55`` which is close to but not exactly "
"equal to the true value of 1/10."
msgstr ""
"只要您停在任何有限的位數，您就只會得到近似值。而現在大多數的計算機中，浮點數"
"是透過二進位分數近似的，其中分子從最高有效位元使開始用 53 個位元表示，分母則"
"是以二為底的指數。在 1/10 的例子中，二進位分數為 ``3602879701896397 / 2 ** "
"55``，而這樣的表示十分地接近，但不完全等同於 1/10 的真正數值。"
#: ../../tutorial/floatingpoint.rst:62
msgid ""
"Many users are not aware of the approximation because of the way values are "
"displayed. Python only prints a decimal approximation to the true decimal "
"value of the binary approximation stored by the machine. On most machines, "
"if Python were to print the true decimal value of the binary approximation "
"stored for 0.1, it would have to display ::"
msgstr ""
"由於數值顯示的方式，很多使用者並沒有發現數值是個近似值。Python 只會印出一個十"
"進位近似值，其近似了儲存在計算機中的二進位近似值的十進位數值。在大多數的計算"
"機中，如果 Python 真的會印出完整的十進位數值，其表示儲存在計算機中的 0.1 的二"
"進位近似值，它將顯示為：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:71
msgid ""
"That is more digits than most people find useful, so Python keeps the number "
"of digits manageable by displaying a rounded value instead ::"
msgstr ""
"這比一般人感到有用的位數還多，所以 Python 將位數保持在可以接受的範圍，只顯示"
"捨入後的數值：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:77
msgid ""
"Just remember, even though the printed result looks like the exact value of "
"1/10, the actual stored value is the nearest representable binary fraction."
msgstr ""
"一定要記住，雖然印出的數字看起來是精準的 1/10，但真正儲存的數值是能表示的二進"
"位分數中，最接近精準數值的數。"
#: ../../tutorial/floatingpoint.rst:80
msgid ""
"Interestingly, there are many different decimal numbers that share the same "
"nearest approximate binary fraction. For example, the numbers ``0.1`` and "
"``0.10000000000000001`` and "
"``0.1000000000000000055511151231257827021181583404541015625`` are all "
"approximated by ``3602879701896397 / 2 ** 55``. Since all of these decimal "
"values share the same approximation, any one of them could be displayed "
"while still preserving the invariant ``eval(repr(x)) == x``."
msgstr ""
"有趣的是，有許多不同的十進位數，共用同一個最接近的二進位近似小數。例如說：數"
"字 ``0.1`` 和 ``0.10000000000000001`` 和 "
"``0.1000000000000000055511151231257827021181583404541015625``，都由 "
"``3602879701896397 / 2 ** 55`` 近似。由於這三個數值共用同一個近似值，任何一個"
"數值都可以被顯示，同時保持 ``eval(repr(x)) == x``。"
#: ../../tutorial/floatingpoint.rst:88
msgid ""
"Historically, the Python prompt and built-in :func:`repr` function would "
"choose the one with 17 significant digits, ``0.10000000000000001``. "
"Starting with Python 3.1, Python (on most systems) is now able to choose the "
"shortest of these and simply display ``0.1``."
msgstr ""
"歷史上，Python 的提示字元 (prompt) 與內建的 :func:`repr` 函式會選擇上段說明中"
"有 17 個有效位元的數：``0.10000000000000001``。從 Python 3.1 版開始，"
"Python（在大部分的系統上）可以選擇其中最短的數並簡單地顯示為 ``0.1``。"
#: ../../tutorial/floatingpoint.rst:93
msgid ""
"Note that this is in the very nature of binary floating-point: this is not a "
"bug in Python, and it is not a bug in your code either. You'll see the same "
"kind of thing in all languages that support your hardware's floating-point "
"arithmetic (although some languages may not *display* the difference by "
"default, or in all output modes)."
msgstr ""
"注意，這是二進位浮點數理所當然的特性，並不是 Python 的錯誤 (bug)，更不是您程"
"式碼的錯誤。只要有程式語言支持硬體的浮點數運算，您將會看到同樣的事情出現在其"
"中（雖然某些程式語言預設不\\ *顯示*\\ 差異，或者預設全部輸出）。"
#: ../../tutorial/floatingpoint.rst:99
msgid ""
"For more pleasant output, you may wish to use string formatting to produce a "
"limited number of significant digits::"
msgstr ""
"為求更優雅的輸出，您可能想要使用字串的格式化 (string formatting) 產生限定的有"
"效位數：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:111
msgid ""
"It's important to realize that this is, in a real sense, an illusion: you're "
"simply rounding the *display* of the true machine value."
msgstr ""
"要了解一件很重要的事，在真正意義上，浮點數的表示是一種幻覺：你基本上在捨入真"
"正機器數值所\\ *展示的值*\\ 。"
#: ../../tutorial/floatingpoint.rst:114
msgid ""
"One illusion may beget another. For example, since 0.1 is not exactly 1/10, "
"summing three values of 0.1 may not yield exactly 0.3, either::"
msgstr ""
"這種幻覺可能會產生下一個幻覺。舉例來說，因為 0.1 不是真正的 1/10，把三個 0.1 "
"的值相加，也不會產生精準的 0.3：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:120
msgid ""
"Also, since the 0.1 cannot get any closer to the exact value of 1/10 and 0.3 "
"cannot get any closer to the exact value of 3/10, then pre-rounding with :"
"func:`round` function cannot help::"
msgstr ""
"同時，因為 0.1 不能再更接近精準的 1/10，還有 0.3 不能再更接近精準的 3/10，預"
"先用 :func:`round` 函式捨入並不會有幫助：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:127
msgid ""
"Though the numbers cannot be made closer to their intended exact values, "
"the :func:`round` function can be useful for post-rounding so that results "
"with inexact values become comparable to one another::"
msgstr ""
"雖然數字不會再更接近他們的精準數值，但 :func:`round` 函式可以對事後的捨入有所"
"幫助，如此一來，不精確的數值就變得可以互相比較：\n"
"\n"
"::"
#: ../../tutorial/floatingpoint.rst:134
msgid ""
"Binary floating-point arithmetic holds many surprises like this. The "
"problem with \"0.1\" is explained in precise detail below, in the "
"\"Representation Error\" section. See `The Perils of Floating Point <http://"
"www.lahey.com/float.htm>`_ for a more complete account of other common "
"surprises."
msgstr ""
"二進位浮點數架構擁有很多這樣的驚喜。底下的「表示法錯誤」章節，詳細的解釋了"
"「0.1」的問題。如果想要其他常見驚喜更完整的描述，可以參考 `The Perils of "
"Floating Point（浮點數的風險） <http://www.lahey.com/float.htm>`_。"
#: ../../tutorial/floatingpoint.rst:139
msgid ""
"As that says near the end, \"there are no easy answers.\" Still, don't be "
"unduly wary of floating-point! The errors in Python float operations are "
"inherited from the floating-point hardware, and on most machines are on the "
"order of no more than 1 part in 2\\*\\*53 per operation. That's more than "
"adequate for most tasks, but you do need to keep in mind that it's not "
"decimal arithmetic and that every float operation can suffer a new rounding "
"error."
msgstr ""
#: ../../tutorial/floatingpoint.rst:146
msgid ""
"While pathological cases do exist, for most casual use of floating-point "
"arithmetic you'll see the result you expect in the end if you simply round "
"the display of your final results to the number of decimal digits you "
"expect. :func:`str` usually suffices, and for finer control see the :meth:"
"`str.format` method's format specifiers in :ref:`formatstrings`."
msgstr ""
#: ../../tutorial/floatingpoint.rst:152
msgid ""
"For use cases which require exact decimal representation, try using the :mod:"
"`decimal` module which implements decimal arithmetic suitable for accounting "
"applications and high-precision applications."
msgstr ""
#: ../../tutorial/floatingpoint.rst:156
msgid ""
"Another form of exact arithmetic is supported by the :mod:`fractions` module "
"which implements arithmetic based on rational numbers (so the numbers like "
"1/3 can be represented exactly)."
msgstr ""
#: ../../tutorial/floatingpoint.rst:160
msgid ""
"If you are a heavy user of floating point operations you should take a look "
"at the Numerical Python package and many other packages for mathematical and "
"statistical operations supplied by the SciPy project. See <https://scipy."
"org>."
msgstr ""
#: ../../tutorial/floatingpoint.rst:164
msgid ""
"Python provides tools that may help on those rare occasions when you really "
"*do* want to know the exact value of a float. The :meth:`float."
"as_integer_ratio` method expresses the value of a float as a fraction::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:173
msgid ""
"Since the ratio is exact, it can be used to losslessly recreate the original "
"value::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:179
msgid ""
"The :meth:`float.hex` method expresses a float in hexadecimal (base 16), "
"again giving the exact value stored by your computer::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:185
msgid ""
"This precise hexadecimal representation can be used to reconstruct the float "
"value exactly::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:191
msgid ""
"Since the representation is exact, it is useful for reliably porting values "
"across different versions of Python (platform independence) and exchanging "
"data with other languages that support the same format (such as Java and "
"C99)."
msgstr ""
#: ../../tutorial/floatingpoint.rst:195
msgid ""
"Another helpful tool is the :func:`math.fsum` function which helps mitigate "
"loss-of-precision during summation. It tracks \"lost digits\" as values are "
"added onto a running total. That can make a difference in overall accuracy "
"so that the errors do not accumulate to the point where they affect the "
"final total:"
msgstr ""
#: ../../tutorial/floatingpoint.rst:209
msgid "Representation Error"
msgstr ""
#: ../../tutorial/floatingpoint.rst:211
msgid ""
"This section explains the \"0.1\" example in detail, and shows how you can "
"perform an exact analysis of cases like this yourself. Basic familiarity "
"with binary floating-point representation is assumed."
msgstr ""
#: ../../tutorial/floatingpoint.rst:215
msgid ""
":dfn:`Representation error` refers to the fact that some (most, actually) "
"decimal fractions cannot be represented exactly as binary (base 2) "
"fractions. This is the chief reason why Python (or Perl, C, C++, Java, "
"Fortran, and many others) often won't display the exact decimal number you "
"expect."
msgstr ""
#: ../../tutorial/floatingpoint.rst:220
msgid ""
"Why is that? 1/10 is not exactly representable as a binary fraction. Almost "
"all machines today (November 2000) use IEEE-754 floating point arithmetic, "
"and almost all platforms map Python floats to IEEE-754 \"double precision"
"\". 754 doubles contain 53 bits of precision, so on input the computer "
"strives to convert 0.1 to the closest fraction it can of the form *J*/2**\\ "
"*N* where *J* is an integer containing exactly 53 bits. Rewriting ::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:229
msgid "as ::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:233
msgid ""
"and recalling that *J* has exactly 53 bits (is ``>= 2**52`` but ``< "
"2**53``), the best value for *N* is 56::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:239
msgid ""
"That is, 56 is the only value for *N* that leaves *J* with exactly 53 bits. "
"The best possible value for *J* is then that quotient rounded::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:246
msgid ""
"Since the remainder is more than half of 10, the best approximation is "
"obtained by rounding up::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:252
msgid ""
"Therefore the best possible approximation to 1/10 in 754 double precision "
"is::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:256
msgid ""
"Dividing both the numerator and denominator by two reduces the fraction to::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:260
msgid ""
"Note that since we rounded up, this is actually a little bit larger than "
"1/10; if we had not rounded up, the quotient would have been a little bit "
"smaller than 1/10. But in no case can it be *exactly* 1/10!"
msgstr ""
#: ../../tutorial/floatingpoint.rst:264
msgid ""
"So the computer never \"sees\" 1/10: what it sees is the exact fraction "
"given above, the best 754 double approximation it can get::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:270
msgid ""
"If we multiply that fraction by 10\\*\\*55, we can see the value out to 55 "
"decimal digits::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:276
msgid ""
"meaning that the exact number stored in the computer is equal to the decimal "
"value 0.1000000000000000055511151231257827021181583404541015625. Instead of "
"displaying the full decimal value, many languages (including older versions "
"of Python), round the result to 17 significant digits::"
msgstr ""
#: ../../tutorial/floatingpoint.rst:284
msgid ""
"The :mod:`fractions` and :mod:`decimal` modules make these calculations "
"easy::"
msgstr ""