# Copyright (C) 2001-2024, Python Software Foundation

# This file is distributed under the same license as the Python package.

#

# Translators:

# yichung279, 2018

# cypevan, 2018

# Adrian Liaw <[email protected]>, 2018

# Steven Hsu <[email protected]>, 2021

msgid ""

msgstr ""

"Project-Id-Version: Python 3.12\n"

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"PO-Revision-Date: 2023-06-22 14:43+0800\n"

"Last-Translator: Matt Wang <[email protected]>\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"

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"Plural-Forms: nplurals=1; plural=0;\n"

"X-Generator: Poedit 3.3.1\n"

#: ../../tutorial/floatingpoint.rst:10

msgid "Floating-Point Arithmetic: Issues and Limitations"

msgstr "浮點數運算：問題與限制"

#: ../../tutorial/floatingpoint.rst:16

msgid ""

"Floating-point numbers are represented in computer hardware as base 2 "

"(binary) fractions. For example, the **decimal** fraction ``0.625`` has "

"value 6/10 + 2/100 + 5/1000, and in the same way the **binary** fraction "

"``0.101`` has value 1/2 + 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 ""

"在計算機架構中，浮點數 (floating-point number) 是以基數為 2（二進位）的小數表"

"示。例如說，在\\ **十進位**\\ 小數中 ``0.625`` 可被分為 6/10 + 2/100 + "

"5/1000，同樣的道理，**二進位**\\ 小數 ``0.101`` 可被分為 1/2 + 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:23

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，你輸入的"

"十進位浮點數只能由實際儲存在計算機中的二進位浮點數近似。"

#: ../../tutorial/floatingpoint.rst:28

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，你可以將其近似為十進位小"

"數： ::"

#: ../../tutorial/floatingpoint.rst:31

msgid "0.3"

msgstr "0.3"

#: ../../tutorial/floatingpoint.rst:33 ../../tutorial/floatingpoint.rst:37

msgid "or, better, ::"

msgstr "或者，更好的近似： ::"

#: ../../tutorial/floatingpoint.rst:35

msgid "0.33"

msgstr "0.33"

#: ../../tutorial/floatingpoint.rst:39

msgid "0.333"

msgstr "0.333"

#: ../../tutorial/floatingpoint.rst:41

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:45

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： ::"

#: ../../tutorial/floatingpoint.rst:49

msgid "0.0001100110011001100110011001100110011001100110011..."

msgstr "0.0001100110011001100110011001100110011001100110011..."

#: ../../tutorial/floatingpoint.rst:51

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 ""

"只要你停在任何有限的位數，你就只會得到近似值。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，其中分子是從最高有效位元開始，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:58

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真正十進位數值。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： ::"

#: ../../tutorial/floatingpoint.rst:64

msgid ""

">>> 0.1\n"

"0.1000000000000000055511151231257827021181583404541015625"

msgstr ""

">>> 0.1\n"

"0.1000000000000000055511151231257827021181583404541015625"

#: ../../tutorial/floatingpoint.rst:67

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："

#: ../../tutorial/floatingpoint.rst:70

msgid ""

">>> 1 / 10\n"

"0.1"

msgstr ""

">>> 1 / 10\n"

"0.1"

#: ../../tutorial/floatingpoint.rst:75

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:78

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分數。例如說：數"

"字 ``0.1`` 和 ``0.10000000000000001`` 和 "

"``0.1000000000000000055511151231257827021181583404541015625``，都由 "

"``3602879701896397 / 2 ** 55`` 近似。由於這三個十進位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:86

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）現在能選擇其中最短的數並簡單地顯示為 ``0.1``。"

#: ../../tutorial/floatingpoint.rst:91

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)，更不是你程"

"式碼的錯誤。只要程式語言有支援硬體的浮點數運算，你將會看到同樣的事情出現在其"

"中（雖然有些程式語言預設不會\\ *顯示*\\ 該差異，有些甚至是在所有的輸出模式中"

"都不會顯示。）"

#: ../../tutorial/floatingpoint.rst:97

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，你可能想要使用字串的格式化 (string formatting) 產生限定的有"

"效位數："

#: ../../tutorial/floatingpoint.rst:100

msgid ""

">>> format(math.pi, '.12g') # give 12 significant digits\n"

"'3.14159265359'\n"

"\n"

">>> format(math.pi, '.2f') # give 2 digits after the point\n"

"'3.14'\n"

"\n"

">>> repr(math.pi)\n"

"'3.141592653589793'"

msgstr ""

#: ../../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："

#: ../../tutorial/floatingpoint.rst:117

msgid ""

">>> 0.1 + 0.1 + 0.1 == 0.3\n"

"False"

msgstr ""

">>> 0.1 + 0.1 + 0.1 == 0.3\n"

"False"

#: ../../tutorial/floatingpoint.rst:122

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："

#: ../../tutorial/floatingpoint.rst:126

msgid ""

">>> round(0.1, 1) + round(0.1, 1) + round(0.1, 1) == round(0.3, 1)\n"

"False"

msgstr ""

">>> round(0.1, 1) + round(0.1, 1) + round(0.1, 1) == round(0.3, 1)\n"

"False"

#: ../../tutorial/floatingpoint.rst:131

msgid ""

"Though the numbers cannot be made closer to their intended exact values, "

"the :func:`math.isclose` function can be useful for comparing inexact values:"

msgstr ""

"雖然數字不會再更接近他們的精準數值，但 :func:`math.isclose` 函式可以用來比較"

"不精確的值： ::"

#: ../../tutorial/floatingpoint.rst:134

msgid ""

">>> math.isclose(0.1 + 0.1 + 0.1, 0.3)\n"

"True"

msgstr ""

">>> math.isclose(0.1 + 0.1 + 0.1, 0.3)\n"

"True"

#: ../../tutorial/floatingpoint.rst:139

msgid ""

"Alternatively, the :func:`round` function can be used to compare rough "

"approximations:"

msgstr "或者，可以使用 :func:`round` 函式來比較粗略的近似值："

#: ../../tutorial/floatingpoint.rst:142

msgid ""

">>> round(math.pi, ndigits=2) == round(22 / 7, ndigits=2)\n"

"True"

msgstr ""

">>> round(math.pi, ndigits=2) == round(22 / 7, ndigits=2)\n"

"True"

#: ../../tutorial/floatingpoint.rst:147

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 `Examples of Floating Point Problems "

"<https://jvns.ca/blog/2023/01/13/examples-of-floating-point-problems/>`_ for "

"a pleasant summary of how binary floating point works and the kinds of "

"problems commonly encountered in practice. Also see `The Perils of Floating "

"Point <http://www.indowsway.com/floatingpoint.htm>`_ for a more complete "

"account of other common surprises."

msgstr ""

"二進位浮點數架構下還有很多這樣的意外。底下的「表示法誤差 (Representation "

"Error)」章節，詳細地解釋了「0.1」的問題。`Examples of Floating Point Problems"

"（浮點數問題範例） <https://jvns.ca/blog/2023/01/13/examples-of-floating-"

"point-problems/>`_\\ 一文提供了二進位浮點數的作用方式與現實中這類常見問題的摘"

"錄。如果想要其他常見問題的更完整描述，可以參考 `The Perils of Floating Point"

"（浮點數的風險） <http://www.indowsway.com/floatingpoint.htm>`_。"

#: ../../tutorial/floatingpoint.rst:156

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 ""

"如同上文的末段所述，「這個問題並沒有簡單的答案。」不過，也不必對浮點過度擔"

"心！Python 的 float（浮點數）運算中的誤差，是從浮點硬體繼承而來，而在大多數的"

"計算機上，每次運算的誤差範圍不會大於 2\\*\\*53 分之 1。這對大多數的任務來說已"

"經相當足夠，但你需要記住，它並非十進位運算，且每一次 float 運算都可能會承受新"

"的捨入誤差。"

#: ../../tutorial/floatingpoint.rst:163

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 ""

"雖然浮點運算確實存在一些問題，但在一般情況下，如果你只是把最終結果的顯示值，"

"以十進位方式捨入至預期的位數，那麼仍會得到你預期的結果。:func:`str` 通常就能"

"滿足要求，而若想要更細緻的控制，可參閱\\ :ref:`formatstrings`\\ 中關於 :meth:"

"`str.format` method（方法）的格式規範。"

#: ../../tutorial/floatingpoint.rst:169

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 ""

"對於需要精準十進位表示法的用例，可以試著用 :mod:`decimal` 模組，它可實作適用"

"於會計應用程式與高精確度應用程式的十進位運算。"

#: ../../tutorial/floatingpoint.rst:173

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 ""

"另一種支援精準運算的格式為 :mod:`fractions` 模組，該模組基於有理數來實作運算"

"（因此可以精確表示像 1/3 這樣的數字）。"

#: ../../tutorial/floatingpoint.rst:177

msgid ""

"If you are a heavy user of floating-point operations you should take a look "

"at the NumPy package and many other packages for mathematical and "

"statistical operations supplied by the SciPy project. See <https://scipy."

"org>."

msgstr ""

"如果你是浮點運算的重度使用者，你應該看一下 NumPy 套件，以及由 SciPy 專案提供"

"的許多用於數學和統計學運算的其他套件。請參閱 <https://scipy.org>。"

#: ../../tutorial/floatingpoint.rst:181

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 ""

"在罕見情況下，當你\\ *真的*\\ 想知道一個 float 的精準值，Python 提供的工具可"

"協助達成。:meth:`float.as_integer_ratio` method 可將一個 float 的值表示為分"

"數："

#: ../../tutorial/floatingpoint.rst:186

msgid ""

">>> x = 3.14159\n"

">>> x.as_integer_ratio()\n"

"(3537115888337719, 1125899906842624)"

msgstr ""

">>> x = 3.14159\n"

">>> x.as_integer_ratio()\n"

"(3537115888337719, 1125899906842624)"

#: ../../tutorial/floatingpoint.rst:192

msgid ""

"Since the ratio is exact, it can be used to losslessly recreate the original "

"value:"

msgstr "由於該比率是精準的，它可無損地再現該原始值："

#: ../../tutorial/floatingpoint.rst:195

msgid ""

">>> x == 3537115888337719 / 1125899906842624\n"

"True"

msgstr ""

">>> x == 3537115888337719 / 1125899906842624\n"

"True"

#: ../../tutorial/floatingpoint.rst:200

msgid ""

"The :meth:`float.hex` method expresses a float in hexadecimal (base 16), "

"again giving the exact value stored by your computer:"

msgstr ""

":meth:`float.hex` method 以十六進位（基數為 16）表示 float，一樣可以給出你的"

"電腦所儲存的精準值："

#: ../../tutorial/floatingpoint.rst:203

msgid ""

">>> x.hex()\n"

"'0x1.921f9f01b866ep+1'"

msgstr ""

">>> x.hex()\n"

"'0x1.921f9f01b866ep+1'"

#: ../../tutorial/floatingpoint.rst:208

msgid ""

"This precise hexadecimal representation can be used to reconstruct the float "

"value exactly:"

msgstr "這種精確的十六進位表示法可用於精準地重建 float 值："

#: ../../tutorial/floatingpoint.rst:211

msgid ""

">>> x == float.fromhex('0x1.921f9f01b866ep+1')\n"

"True"

msgstr ""

">>> x == float.fromhex('0x1.921f9f01b866ep+1')\n"

"True"

#: ../../tutorial/floatingpoint.rst:216

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 ""

"由於該表示法是精準的，因此適用於在不同版本的 Python 之間可靠地傳送數值（獨立"

"於系統平台），並與支援相同格式的其他語言（如 JAVA 和 C99）交換資料。"

#: ../../tutorial/floatingpoint.rst:220

msgid ""

"Another helpful tool is the :func:`sum` function which helps mitigate loss-"

"of-precision during summation. It uses extended precision for intermediate "

"rounding steps 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 ""

"另一個有用的工具是 :func:`sum` 函式，能在計算總和時幫忙減少精確度的損失。當數"

"值被加到運行中的總計值時，它會追蹤「失去的位數 (lost digits)」。這可以明顯改"

"善總體準確度 (overall accuracy)，使得誤差不至於累積到影響最終總計值的程度："

#: ../../tutorial/floatingpoint.rst:226

msgid ""

">>> 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 == 1.0\n"

"False\n"

">>> sum([0.1] * 10) == 1.0\n"

"True"

msgstr ""

">>> 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 == 1.0\n"

"False\n"

">>> sum([0.1] * 10) == 1.0\n"

"True"

#: ../../tutorial/floatingpoint.rst:233

msgid ""

"The :func:`math.fsum` goes further and tracks all of the \"lost digits\" as "

"values are added onto a running total so that the result has only a single "

"rounding. This is slower than :func:`sum` but will be more accurate in "

"uncommon cases where large magnitude inputs mostly cancel each other out "

"leaving a final sum near zero:"

msgstr ""

":func:`math.fsum` 更進一步將數值被加到運行總計中的總計時追蹤所有「失去的位"

"數」，以便結果僅有一次捨入。這比 :func:`sum` 慢，但在不常見的情況下會更準確，"

"在這種情況下，大量輸入大多相互抵消，最終的總和會接近於零："

#: ../../tutorial/floatingpoint.rst:239

msgid ""

">>> arr = [-0.10430216751806065, -266310978.67179024, 143401161448607.16,\n"

"... -143401161400469.7, 266262841.31058735, -0.003244936839808227]\n"

">>> float(sum(map(Fraction, arr))) # Exact summation with single rounding\n"

"8.042173697819788e-13\n"

">>> math.fsum(arr) # Single rounding\n"

"8.042173697819788e-13\n"

">>> sum(arr) # Multiple roundings in extended "

"precision\n"

"8.042178034628478e-13\n"

">>> total = 0.0\n"

">>> for x in arr:\n"

"... total += x # Multiple roundings in standard "

"precision\n"

"...\n"

">>> total # Straight addition has no correct "

"digits!\n"

"-0.0051575902860057365"

msgstr ""

#: ../../tutorial/floatingpoint.rst:260

msgid "Representation Error"

msgstr "表示法誤差 (Representation Error)"

#: ../../tutorial/floatingpoint.rst:262

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 ""

"本節將會詳細解釋「0.1」的例子，並說明你自己要如何對類似案例執行精準的分析。 "

"以下假設你對二進位浮點表示法已有基本程度的熟悉。"

#: ../../tutorial/floatingpoint.rst:266

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 ""

":dfn:`Representation error`\\ （表示法誤差）是指在真實情況中，有些（實際上是"

"大多數）十進位小數並不能精準地以二進位（基數為 2）小數表示。這是 Python（或 "

"Perl、C、C++、JAVA、Fortran 和其他許多）通常不會顯示你期望的精準十進位數字的"

"主要原因。"

#: ../../tutorial/floatingpoint.rst:271

msgid ""

"Why is that? 1/10 is not exactly representable as a binary fraction. Since "

"at least 2000, almost all machines use IEEE 754 binary floating-point "

"arithmetic, and almost all platforms map Python floats to IEEE 754 binary64 "

"\"double precision\" values. IEEE 754 binary64 values 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 ""

"為什麼呢？因為 1/10 無法精準地以一個二進位小數來表示。至少自從 2000 年以來，"

"幾乎所有的計算機皆使用 IEEE-754 浮點數運算標準，並且幾乎所有的平台都以 IEEE "

"754 binary64 標準中的「雙精度 (double precision)」來作為 Python 的 float。"

"IEEE 754 binary64 的值包含 53 位元的精度，所以在輸入時，電腦會努力把 0.1 轉換"

"到最接近的分數，以 *J*/2**\\ *N* 的形式表示，此處 *J* 是一個正好包含 53 位元"

"的整數。可以將： ::"

#: ../../tutorial/floatingpoint.rst:280

msgid "1 / 10 ~= J / (2**N)"

msgstr "1 / 10 ~= J / (2**N)"

#: ../../tutorial/floatingpoint.rst:282

msgid "as ::"

msgstr "重寫為： ::"

#: ../../tutorial/floatingpoint.rst:284

msgid "J ~= 2**N / 10"

msgstr "J ~= 2**N / 10"

#: ../../tutorial/floatingpoint.rst:286

msgid ""

"and recalling that *J* has exactly 53 bits (is ``>= 2**52`` but ``< "

"2**53``), the best value for *N* is 56:"

msgstr ""

"而前面提到 *J* 有精準的 53 位元（即 ``>= 2**52`` 但 ``< 2**53``），所以 *N* "

"的最佳數值是 56："

#: ../../tutorial/floatingpoint.rst:289

msgid ""

">>> 2**52 <= 2**56 // 10 < 2**53\n"

"True"

msgstr ""

">>> 2**52 <= 2**56 // 10 < 2**53\n"

"True"

#: ../../tutorial/floatingpoint.rst:294

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 ""

"意即，要使 *J* 正好有 53 位元，則 56 會是 *N* 的唯一值。而 *J* 最有可能的數值"

"就是經過捨入後的該商數："

#: ../../tutorial/floatingpoint.rst:297

msgid ""

">>> q, r = divmod(2**56, 10)\n"

">>> r\n"

"6"

msgstr ""

">>> q, r = divmod(2**56, 10)\n"

">>> r\n"

"6"

#: ../../tutorial/floatingpoint.rst:303

msgid ""

"Since the remainder is more than half of 10, the best approximation is "

"obtained by rounding up:"

msgstr "由於餘數超過 10 的一半，所以最佳的近似值是透過進位而得："

#: ../../tutorial/floatingpoint.rst:306

msgid ""

">>> q+1\n"

"7205759403792794"

msgstr ""

">>> q+1\n"

"7205759403792794"

#: ../../tutorial/floatingpoint.rst:313

msgid ""

"Therefore the best possible approximation to 1/10 in IEEE 754 double "

"precision is::"

msgstr "所以，在 IEEE 754 雙精度下，1/10 的最佳近似值是： ::"

#: ../../tutorial/floatingpoint.rst:316

msgid "7205759403792794 / 2 ** 56"

msgstr "7205759403792794 / 2 ** 56"

#: ../../tutorial/floatingpoint.rst:318

msgid ""

"Dividing both the numerator and denominator by two reduces the fraction to::"

msgstr "將分子和分母同除以二，會約分為： ::"

#: ../../tutorial/floatingpoint.rst:320

msgid "3602879701896397 / 2 ** 55"

msgstr "3602879701896397 / 2 ** 55"

#: ../../tutorial/floatingpoint.rst:322

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 ""

"請注意，由於我們有進位，所以這實際上比 1/10 大了一點；如果我們沒有進位，商數"

"將會有點小於 1/10。但在任何情況下都不可能是\\ *精準的* 1/10！"

#: ../../tutorial/floatingpoint.rst:326

msgid ""

"So the computer never \"sees\" 1/10: what it sees is the exact fraction "

"given above, the best IEEE 754 double approximation it can get:"

msgstr ""

"所以電腦從來沒有「看到」1/10：它看到的是上述的精準分數，也就是它能得到的 "

"IEEE 754 double 最佳近似值： ::"

#: ../../tutorial/floatingpoint.rst:329

msgid ""

">>> 0.1 * 2 ** 55\n"

"3602879701896397.0"

msgstr ""

">>> 0.1 * 2 ** 55\n"

"3602879701896397.0"

#: ../../tutorial/floatingpoint.rst:334

msgid ""

"If we multiply that fraction by 10\\*\\*55, we can see the value out to 55 "

"decimal digits:"

msgstr "如果將該分數乘以 10\\*\\*55，則可以看到該值以 55 個十進位數字顯示："

#: ../../tutorial/floatingpoint.rst:337

msgid ""

">>> 3602879701896397 * 10 ** 55 // 2 ** 55\n"

"1000000000000000055511151231257827021181583404541015625"

msgstr ""

">>> 3602879701896397 * 10 ** 55 // 2 ** 55\n"

"1000000000000000055511151231257827021181583404541015625"

#: ../../tutorial/floatingpoint.rst:342

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 ""

"這表示儲存在電腦中的精準數值等於十進位值 "

"0.1000000000000000055511151231257827021181583404541015625。與其顯示完整的十進"

"位數值，許多語言（包括 Python 的舊版本）選擇將結果捨入至 17 個有效位數："

#: ../../tutorial/floatingpoint.rst:347

msgid ""

">>> format(0.1, '.17f')\n"

"'0.10000000000000001'"

msgstr ""

">>> format(0.1, '.17f')\n"

"'0.10000000000000001'"

#: ../../tutorial/floatingpoint.rst:352

msgid ""

"The :mod:`fractions` and :mod:`decimal` modules make these calculations easy:"

msgstr ":mod:`fractions` 與 :mod:`decimal` 模組能使這些計算變得容易："

#: ../../tutorial/floatingpoint.rst:355

msgid ""

">>> from decimal import Decimal\n"

">>> from fractions import Fraction\n"

"\n"

">>> Fraction.from_float(0.1)\n"

"Fraction(3602879701896397, 36028797018963968)\n"

"\n"

">>> (0.1).as_integer_ratio()\n"

"(3602879701896397, 36028797018963968)\n"

"\n"

">>> Decimal.from_float(0.1)\n"

"Decimal('0.1000000000000000055511151231257827021181583404541015625')\n"

"\n"

">>> format(Decimal.from_float(0.1), '.17')\n"

"'0.10000000000000001'"

msgstr ""

">>> from decimal import Decimal\n"

">>> from fractions import Fraction\n"

"\n"

">>> Fraction.from_float(0.1)\n"

"Fraction(3602879701896397, 36028797018963968)\n"

"\n"

">>> (0.1).as_integer_ratio()\n"

"(3602879701896397, 36028797018963968)\n"

"\n"

">>> Decimal.from_float(0.1)\n"

"Decimal('0.1000000000000000055511151231257827021181583404541015625')\n"

"\n"

">>> format(Decimal.from_float(0.1), '.17')\n"

"'0.10000000000000001'"