From 338331f69da3d8c1b004362ab9ef7ce51fd8b6d4 Mon Sep 17 00:00:00 2001
From: Enkai <enkaihuang2021@gmail.com>
Date: Mon, 11 Mar 2024 21:19:57 +0800
Subject: [PATCH 1/4] Translate cmath

---
 library/cmath.po | 120 +++++++++++++++++++++++++++++++++++------------
 1 file changed, 90 insertions(+), 30 deletions(-)

diff --git a/library/cmath.po b/library/cmath.po
index f85f0d8f4d..4aa27a9c75 100644
--- a/library/cmath.po
+++ b/library/cmath.po
@@ -8,7 +8,7 @@ msgstr ""
 "Project-Id-Version: Python 3.12\n"
 "Report-Msgid-Bugs-To: \n"
 "POT-Creation-Date: 2023-06-30 15:31+0000\n"
-"PO-Revision-Date: 2018-05-23 14:40+0000\n"
+"PO-Revision-Date: 2024-03-11 20:33+0800\n"
 "Last-Translator: Adrian Liaw <adrianliaw2000@gmail.com>\n"
 "Language-Team: Chinese - TAIWAN (https://github.com/python/python-docs-zh-"
 "tw)\n"
@@ -17,10 +17,11 @@ msgstr ""
 "Content-Type: text/plain; charset=UTF-8\n"
 "Content-Transfer-Encoding: 8bit\n"
 "Plural-Forms: nplurals=1; plural=0;\n"
+"X-Generator: Poedit 3.4.2\n"
 
 #: ../../library/cmath.rst:2
 msgid ":mod:`cmath` --- Mathematical functions for complex numbers"
-msgstr ""
+msgstr ":mod:`cmath` --- 複數的數學函式"
 
 #: ../../library/cmath.rst:9
 msgid ""
@@ -32,6 +33,10 @@ msgid ""
 "floating-point number, respectively, and the function is then applied to the "
 "result of the conversion."
 msgstr ""
+"本模組提供一些適用於複數的數學函式。本模組中的函式接受整數、浮點數或複數作為"
+"引數。它們也接受任何具有 :meth:`~object.__complex__` 或 :meth:`~object."
+"__float__` 方法的 Python 物件：這些方法分別用於將物件轉換為複數或浮點數，然後"
+"再將函式應用於轉換後的結果。"
 
 #: ../../library/cmath.rst:18
 msgid ""
@@ -43,6 +48,11 @@ msgid ""
 "axis we look at the sign of the imaginary part, while for a branch cut along "
 "the imaginary axis we look at the sign of the real part."
 msgstr ""
+"對於涉及分枝切割 (branch cut) 的函式，我們面臨的問題是決定如何定義在切割本身"
+"上的這些函式。遵循 Kahan 的論文 \"Branch cuts for complex elementary "
+"functions\" ，以及 C99 的附錄 G 和後來的 C 標準，我們使用零符號來區分分枝切割"
+"的兩側：對於沿著（一部分）實數軸的分枝切割，我們查看虛部的符號，而對於沿虛軸"
+"的分枝切割，我們則查看實部的符號。"
 
 #: ../../library/cmath.rst:26
 msgid ""
@@ -51,16 +61,19 @@ msgid ""
 "though it lies *below* the branch cut, and so gives a result on the negative "
 "imaginary axis::"
 msgstr ""
+"例如 :func:`cmath.sqrt` 函式具有一條沿負實軸的分枝切割。 引數 "
+"``complex(-2.0, -0.0)`` 被視為位於分枝切割 *下方* 處理，因此給出的結果在負虛"
+"軸上： ::"
 
 #: ../../library/cmath.rst:34
 msgid ""
 "But an argument of ``complex(-2.0, 0.0)`` is treated as though it lies above "
 "the branch cut::"
-msgstr ""
+msgstr "但是引數 ``complex(-2.0, 0.0)`` 會被當成位於分枝切割上方處理： ::"
 
 #: ../../library/cmath.rst:42
 msgid "Conversions to and from polar coordinates"
-msgstr ""
+msgstr "轉換到極座標和從極座標做轉換"
 
 #: ../../library/cmath.rst:44
 msgid ""
@@ -68,6 +81,8 @@ msgid ""
 "*Cartesian* coordinates.  It is completely determined by its *real part* ``z."
 "real`` and its *imaginary part* ``z.imag``.  In other words::"
 msgstr ""
+"Python 複數 ``z`` 是用 *直角坐標* 或 *笛卡爾坐標* 儲存在內部的。它完全是由其 "
+"*實部* ``z.real``和 *虛部* ``z.imag`` 所決定。換句話說： ::"
 
 #: ../../library/cmath.rst:51
 msgid ""
@@ -78,12 +93,15 @@ msgid ""
 "radians, from the positive x-axis to the line segment that joins the origin "
 "to *z*."
 msgstr ""
+"*極座標* 提供了另一種表示複數的方法。在極座標中，複數 *z* 由絕對值 (modulus) "
+"*r* 和相位角 (phase) *phi* 定義。絕對值 *r* 是從 *z* 到原點的距離，而相位角 "
+"*phi* 是從正 x 軸到連接原點到 *z* 的線段的逆時針角度（以弧度為單位）。"
 
 #: ../../library/cmath.rst:58
 msgid ""
 "The following functions can be used to convert from the native rectangular "
 "coordinates to polar coordinates and back."
-msgstr ""
+msgstr "以下的函式可用於原始直角座標與極座標之間的相互轉換。"
 
 #: ../../library/cmath.rst:63
 msgid ""
@@ -93,6 +111,10 @@ msgid ""
 "along the negative real axis.  The sign of the result is the same as the "
 "sign of ``x.imag``, even when ``x.imag`` is zero::"
 msgstr ""
+"以浮點數的形式回傳 *x* 的相位角（也稱為 *x* 的 *引數* ）。 ``phase(x)`` 等價"
+"於 ``math.atan2(x.imag, x.real)`` 。結果將位於 [-\\ *π*, *π*] 的範圍內，且此"
+"操作的分枝切割將位於負實軸上。結果的符號會與 ``x.imag`` 的符號相同，即使 ``x."
+"imag`` 為零： ::"
 
 #: ../../library/cmath.rst:77
 msgid ""
@@ -100,6 +122,8 @@ msgid ""
 "the built-in :func:`abs` function.  There is no separate :mod:`cmath` module "
 "function for this operation."
 msgstr ""
+"複數 *x* 的絕對值可以使用內建的 :func:`abs` 函式計算。沒有單獨的 :mod:"
+"`cmath` 模組函式適用於此操作。"
 
 #: ../../library/cmath.rst:84
 msgid ""
@@ -107,22 +131,26 @@ msgid ""
 "phi)`` where *r* is the modulus of *x* and phi is the phase of *x*.  "
 "``polar(x)`` is equivalent to ``(abs(x), phase(x))``."
 msgstr ""
+"回傳 *x* 在極座標中的表達方式。回傳一組數對 ``(r, phi)`` ， *r* 是 *x* 的絕對"
+"值， *phi* 是 *x* 的相位角。 ``polar(x)`` 相當於 ``(abs(x), phase(x))`` 。"
 
 #: ../../library/cmath.rst:92
 msgid ""
 "Return the complex number *x* with polar coordinates *r* and *phi*. "
 "Equivalent to ``r * (math.cos(phi) + math.sin(phi)*1j)``."
 msgstr ""
+"透過極座標 *r* 和 *phi* 回傳複數 *x* 。相當於 ``r * (math.cos(phi) + math."
+"sin(phi)*1j)`` 。"
 
 #: ../../library/cmath.rst:97
 msgid "Power and logarithmic functions"
-msgstr ""
+msgstr "冪函數和對數函數"
 
 #: ../../library/cmath.rst:101
 msgid ""
 "Return *e* raised to the power *x*, where *e* is the base of natural "
 "logarithms."
-msgstr ""
+msgstr "回傳 *e* 的 *x* 次方，其中 *e* 是自然對數的底數。"
 
 #: ../../library/cmath.rst:107
 msgid ""
@@ -130,21 +158,23 @@ msgid ""
 "specified, returns the natural logarithm of *x*. There is one branch cut, "
 "from 0 along the negative real axis to -∞."
 msgstr ""
+"回傳 *x* 給定 *base* 的對數。如果未指定 *base* ，則傳回 *x* 的自然對數。存在"
+"一條分枝切割，從 0 沿負實數軸到 -∞ 。"
 
 #: ../../library/cmath.rst:114
 msgid ""
 "Return the base-10 logarithm of *x*. This has the same branch cut as :func:"
 "`log`."
-msgstr ""
+msgstr "回傳 *x* 以 10 為底的對數。它與 :func:`log` 具有相同的分枝切割。"
 
 #: ../../library/cmath.rst:120
 msgid ""
 "Return the square root of *x*. This has the same branch cut as :func:`log`."
-msgstr ""
+msgstr "回傳 *x* 的平方根。它與 :func:`log` 具有相同的分枝切割。"
 
 #: ../../library/cmath.rst:124
 msgid "Trigonometric functions"
-msgstr ""
+msgstr "三角函數"
 
 #: ../../library/cmath.rst:128
 msgid ""
@@ -152,11 +182,13 @@ msgid ""
 "from 1 along the real axis to ∞. The other extends left from -1 along the "
 "real axis to -∞."
 msgstr ""
+"回傳 *x* 的反餘弦值。存在兩條分枝切割：一條是從 1 沿著實數軸向右延伸到 ∞。另"
+"一條從 -1 沿實數軸向左延伸到 -∞。"
 
 #: ../../library/cmath.rst:135
 msgid ""
 "Return the arc sine of *x*. This has the same branch cuts as :func:`acos`."
-msgstr ""
+msgstr "回傳 *x* 的反正弦值。它與 :func:`acos` 具有相同的分枝切割。"
 
 #: ../../library/cmath.rst:140
 msgid ""
@@ -164,28 +196,31 @@ msgid ""
 "``1j`` along the imaginary axis to ``∞j``. The other extends from ``-1j`` "
 "along the imaginary axis to ``-∞j``."
 msgstr ""
+"回傳 *x* 的反正切值。有兩條分枝切割：一條是從 ``1j`` 沿著虛軸延伸到 ``∞j`` 。"
+"另一條從 ``-1j`` 沿著虛軸延伸到 ``-∞j`` 。"
 
 #: ../../library/cmath.rst:147
 msgid "Return the cosine of *x*."
-msgstr ""
+msgstr "回傳 *x* 的餘弦值。"
 
 #: ../../library/cmath.rst:152
 msgid "Return the sine of *x*."
-msgstr ""
+msgstr "回傳 *x* 的正弦值。"
 
 #: ../../library/cmath.rst:157
 msgid "Return the tangent of *x*."
-msgstr ""
+msgstr "回傳 *x* 的正切值。"
 
 #: ../../library/cmath.rst:161
 msgid "Hyperbolic functions"
-msgstr ""
+msgstr "雙曲函數"
 
 #: ../../library/cmath.rst:165
 msgid ""
 "Return the inverse hyperbolic cosine of *x*. There is one branch cut, "
 "extending left from 1 along the real axis to -∞."
 msgstr ""
+"回傳 *x* 的反雙曲餘弦值。存在一條分枝切割，從 1 沿實數軸向左延伸到 -∞。"
 
 #: ../../library/cmath.rst:171
 msgid ""
@@ -193,6 +228,8 @@ msgid ""
 "extends from ``1j`` along the imaginary axis to ``∞j``.  The other extends "
 "from ``-1j`` along the imaginary axis to ``-∞j``."
 msgstr ""
+"回傳 *x* 的反雙曲正弦值。存在兩條分枝切割：一條是從 ``1j`` 沿著虛軸延伸到 "
+"``∞j`` 。另一條從 ``-1j`` 沿著虛軸延伸到 ``-∞j`` 。"
 
 #: ../../library/cmath.rst:178
 msgid ""
@@ -200,52 +237,56 @@ msgid ""
 "extends from ``1`` along the real axis to ``∞``. The other extends from "
 "``-1`` along the real axis to ``-∞``."
 msgstr ""
+"回傳 *x* 的反雙曲正切值。存在兩條分枝切割：一條是從 ``1`` 沿著實數軸延伸到 "
+"``∞`` 。另一條從 ``-1`` 沿著實數軸延伸到 ``-∞`` 。"
 
 #: ../../library/cmath.rst:185
 msgid "Return the hyperbolic cosine of *x*."
-msgstr ""
+msgstr "回傳 *x* 的反雙曲餘弦值。"
 
 #: ../../library/cmath.rst:190
 msgid "Return the hyperbolic sine of *x*."
-msgstr ""
+msgstr "回傳 *x* 的反雙曲正弦值。"
 
 #: ../../library/cmath.rst:195
 msgid "Return the hyperbolic tangent of *x*."
-msgstr ""
+msgstr "回傳 *x* 的反雙曲正切值。"
 
 #: ../../library/cmath.rst:199
 msgid "Classification functions"
-msgstr ""
+msgstr "分類函式"
 
 #: ../../library/cmath.rst:203
 msgid ""
 "Return ``True`` if both the real and imaginary parts of *x* are finite, and "
 "``False`` otherwise."
 msgstr ""
+"如果 *x* 的實部和虛部都是有限的，則回傳 ``True`` ，否則回傳 ``False`` 。"
 
 #: ../../library/cmath.rst:211
 msgid ""
 "Return ``True`` if either the real or the imaginary part of *x* is an "
 "infinity, and ``False`` otherwise."
-msgstr ""
+msgstr "如果 *x* 的實部或虛部是無窮大，則回傳 ``True`` ，否則回傳 ``False`` 。"
 
 #: ../../library/cmath.rst:217
 msgid ""
 "Return ``True`` if either the real or the imaginary part of *x* is a NaN, "
 "and ``False`` otherwise."
-msgstr ""
+msgstr "如果 *x* 的實部或虛部為 NaN，則回傳 ``True`` ，否則回傳 ``False`` 。"
 
 #: ../../library/cmath.rst:223
 msgid ""
 "Return ``True`` if the values *a* and *b* are close to each other and "
 "``False`` otherwise."
-msgstr ""
+msgstr "如果 *a* 和 *b* 的值相互接近，則回傳 ``True`` ，否則回傳 ``False`` 。"
 
 #: ../../library/cmath.rst:226
 msgid ""
 "Whether or not two values are considered close is determined according to "
 "given absolute and relative tolerances."
 msgstr ""
+"兩個值是否被認為相互接近是由給定的絕對和相對容許偏差 (tolerance) 所決定的。"
 
 #: ../../library/cmath.rst:229
 msgid ""
@@ -255,18 +296,26 @@ msgid ""
 "tolerance is ``1e-09``, which assures that the two values are the same "
 "within about 9 decimal digits.  *rel_tol* must be greater than zero."
 msgstr ""
+"*rel_tol* 是相對容許偏差 -- 它是 *a* 和 *b* 之間的最大容許偏差值，相對於 *a* "
+"或 *b* 的較大絕對值。例如，要設定 5% 的容許偏差，請傳遞 ``rel_tol=0.05`` 。預"
+"設容許偏差是 ``1e-09`` ，它確保兩個值在大約 9 位十進制數字內相同。 *rel_tol* "
+"必須大於零。"
 
 #: ../../library/cmath.rst:235
 msgid ""
 "*abs_tol* is the minimum absolute tolerance -- useful for comparisons near "
 "zero. *abs_tol* must be at least zero."
 msgstr ""
+"*abs_tol* 是最小絕對容許偏差 -- 對於接近零的比較很有用。 *abs_tol* 必須至少為"
+"零。"
 
 #: ../../library/cmath.rst:238
 msgid ""
 "If no errors occur, the result will be: ``abs(a-b) <= max(rel_tol * "
 "max(abs(a), abs(b)), abs_tol)``."
 msgstr ""
+"如果未發生錯誤，結果將為： ``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), "
+"abs_tol)`` 。"
 
 #: ../../library/cmath.rst:241
 msgid ""
@@ -275,10 +324,13 @@ msgid ""
 "close to any other value, including ``NaN``.  ``inf`` and ``-inf`` are only "
 "considered close to themselves."
 msgstr ""
+"IEEE 754 特殊值 ``NaN`` 、 ``inf`` 和 ``-inf`` 將會根據 IEEE 規則進行處理。具"
+"體來說， ``NaN`` 不被認為接近任何其他值，包括 ``NaN`` 。 ``inf`` 和 ``-inf`` "
+"只被認為是接近它們自己的。"
 
 #: ../../library/cmath.rst:250
 msgid ":pep:`485` -- A function for testing approximate equality"
-msgstr ""
+msgstr ":pep:`485` -- 一個用來測試近似相等的函式"
 
 #: ../../library/cmath.rst:254
 msgid "Constants"
@@ -286,37 +338,37 @@ msgstr "常數"
 
 #: ../../library/cmath.rst:258
 msgid "The mathematical constant *π*, as a float."
-msgstr ""
+msgstr "數學常數 *π* ，作為一個浮點數。"
 
 #: ../../library/cmath.rst:263
 msgid "The mathematical constant *e*, as a float."
-msgstr ""
+msgstr "數學常數 *e* ，作為一個浮點數。"
 
 #: ../../library/cmath.rst:268
 msgid "The mathematical constant *τ*, as a float."
-msgstr ""
+msgstr "數學常數 *τ* ，作為一個浮點數。"
 
 #: ../../library/cmath.rst:275
 msgid "Floating-point positive infinity. Equivalent to ``float('inf')``."
-msgstr ""
+msgstr "正無窮大的浮點數。相當於 ``float('inf')`` 。"
 
 #: ../../library/cmath.rst:282
 msgid ""
 "Complex number with zero real part and positive infinity imaginary part. "
 "Equivalent to ``complex(0.0, float('inf'))``."
-msgstr ""
+msgstr "零為實部和正無窮為虛部的複數。相當於 ``complex(0.0, float('inf'))`` 。"
 
 #: ../../library/cmath.rst:290
 msgid ""
 "A floating-point \"not a number\" (NaN) value.  Equivalent to "
 "``float('nan')``."
-msgstr ""
+msgstr "浮點\\\"非數字\\\" (NaN) 值。相當於 ``float('nan')`` 。"
 
 #: ../../library/cmath.rst:298
 msgid ""
 "Complex number with zero real part and NaN imaginary part. Equivalent to "
 "``complex(0.0, float('nan'))``."
-msgstr ""
+msgstr "零為實部和 NaN 為虛部的複數。相當於 ``complex(0.0, float('nan'))`` 。"
 
 #: ../../library/cmath.rst:306
 msgid ""
@@ -329,6 +381,10 @@ msgid ""
 "as a real number (in which case the complex number has an imaginary part of "
 "zero)."
 msgstr ""
+"請注意，函式的選擇與模組 :mod:`math` 的類似，但並不完全相同。擁有兩個模組的原"
+"因是有些用戶對複數不感興趣，甚至根本就不知道它們是什麼。他們寧願讓 ``math."
+"sqrt(-1)`` 引發異常，也不願它回傳複數。另請注意， :mod:`cmath` 中所定義的函式"
+"始終都會回傳複數，即使答案可以表示為實數（在這種情況下，複數的虛部為零）。"
 
 #: ../../library/cmath.rst:314
 msgid ""
@@ -340,6 +396,10 @@ msgid ""
 "choice of branch cuts for numerical purposes, a good reference should be the "
 "following:"
 msgstr ""
+"關於分枝切割的註釋：它們是沿著給定的不連續函式的曲線。它們是許多複變函數的必"
+"要特徵。假設您需要使用複變函數進行計算，您將會了解分枝切割的概念。請參閱幾乎"
+"所有關於複變函數的（不是太初級的）書籍以獲得啟發。對於如何正確地基於數值目的"
+"選擇分枝切割的相關訊息，以下內容應該是一個很好的參考："
 
 #: ../../library/cmath.rst:324
 msgid ""

From 28e624d720cfe1039ef6126a7fcc2ed68cf5b685 Mon Sep 17 00:00:00 2001
From: Enkai Huang <enkaihuang2021@gmail.com>
Date: Mon, 11 Mar 2024 13:28:08 +0000
Subject: [PATCH 2/4] fix typo

---
 library/cmath.po | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/library/cmath.po b/library/cmath.po
index 4aa27a9c75..5a084a0d8a 100644
--- a/library/cmath.po
+++ b/library/cmath.po
@@ -9,7 +9,7 @@ msgstr ""
 "Report-Msgid-Bugs-To: \n"
 "POT-Creation-Date: 2023-06-30 15:31+0000\n"
 "PO-Revision-Date: 2024-03-11 20:33+0800\n"
-"Last-Translator: Adrian Liaw <adrianliaw2000@gmail.com>\n"
+"Last-Translator: Enkai Huang <enkaihuang2021@gmail.com>\n"
 "Language-Team: Chinese - TAIWAN (https://github.com/python/python-docs-zh-"
 "tw)\n"
 "Language: zh_TW\n"
@@ -82,7 +82,7 @@ msgid ""
 "real`` and its *imaginary part* ``z.imag``.  In other words::"
 msgstr ""
 "Python 複數 ``z`` 是用 *直角坐標* 或 *笛卡爾坐標* 儲存在內部的。它完全是由其 "
-"*實部* ``z.real``和 *虛部* ``z.imag`` 所決定。換句話說： ::"
+"*實部* ``z.real`` 和 *虛部* ``z.imag`` 所決定。換句話說： ::"
 
 #: ../../library/cmath.rst:51
 msgid ""

From f98f1dc04429ab1463dd214cb63b1d7c6c917e38 Mon Sep 17 00:00:00 2001
From: Enkai <enkaihuang2021@gmail.com>
Date: Thu, 14 Mar 2024 10:24:28 +0800
Subject: [PATCH 3/4] update cmath

---
 library/cmath.po | 121 +++++++++++++++++++++++++++++++++++------------
 1 file changed, 90 insertions(+), 31 deletions(-)

diff --git a/library/cmath.po b/library/cmath.po
index f85f0d8f4d..219d52293f 100644
--- a/library/cmath.po
+++ b/library/cmath.po
@@ -8,7 +8,7 @@ msgstr ""
 "Project-Id-Version: Python 3.12\n"
 "Report-Msgid-Bugs-To: \n"
 "POT-Creation-Date: 2023-06-30 15:31+0000\n"
-"PO-Revision-Date: 2018-05-23 14:40+0000\n"
+"PO-Revision-Date: 2024-03-14 09:26+0800\n"
 "Last-Translator: Adrian Liaw <adrianliaw2000@gmail.com>\n"
 "Language-Team: Chinese - TAIWAN (https://github.com/python/python-docs-zh-"
 "tw)\n"
@@ -17,10 +17,11 @@ msgstr ""
 "Content-Type: text/plain; charset=UTF-8\n"
 "Content-Transfer-Encoding: 8bit\n"
 "Plural-Forms: nplurals=1; plural=0;\n"
+"X-Generator: Poedit 3.4.2\n"
 
 #: ../../library/cmath.rst:2
 msgid ":mod:`cmath` --- Mathematical functions for complex numbers"
-msgstr ""
+msgstr ":mod:`cmath` --- 複數的數學函式"
 
 #: ../../library/cmath.rst:9
 msgid ""
@@ -32,6 +33,10 @@ msgid ""
 "floating-point number, respectively, and the function is then applied to the "
 "result of the conversion."
 msgstr ""
+"本模組提供一些適用於複數的數學函式。本模組中的函式接受整數、浮點數或複數作為"
+"引數。它們也接受任何具有 :meth:`~object.__complex__` 或 :meth:`~object."
+"__float__` 方法的 Python 物件：這些方法分別用於將物件轉換為複數或浮點數，然後"
+"再將函式應用於轉換後的結果。"
 
 #: ../../library/cmath.rst:18
 msgid ""
@@ -43,6 +48,11 @@ msgid ""
 "axis we look at the sign of the imaginary part, while for a branch cut along "
 "the imaginary axis we look at the sign of the real part."
 msgstr ""
+"對於涉及分枝切割 (branch cut) 的函式，我們面臨的問題是決定如何定義在切割本身"
+"上的這些函式。遵循 Kahan 的論文 \"Branch cuts for complex elementary "
+"functions\"，以及 C99 的附錄 G 和後來的 C 標準，我們使用零符號來區分分枝切割"
+"的兩側：對於沿著（一部分）實數軸的分枝切割，我們查看虛部的符號，而對於沿虛軸"
+"的分枝切割，我們則查看實部的符號。"
 
 #: ../../library/cmath.rst:26
 msgid ""
@@ -51,16 +61,19 @@ msgid ""
 "though it lies *below* the branch cut, and so gives a result on the negative "
 "imaginary axis::"
 msgstr ""
+"例如 :func:`cmath.sqrt` 函式具有一條沿負實軸的分枝切割。 引數 "
+"``complex(-2.0, -0.0)`` 被視為位於分枝切割 *下方* 處理，因此給出的結果在負虛"
+"軸上： ::"
 
 #: ../../library/cmath.rst:34
 msgid ""
 "But an argument of ``complex(-2.0, 0.0)`` is treated as though it lies above "
 "the branch cut::"
-msgstr ""
+msgstr "但是引數 ``complex(-2.0, 0.0)`` 會被當成位於分枝切割上方處理： ::"
 
 #: ../../library/cmath.rst:42
 msgid "Conversions to and from polar coordinates"
-msgstr ""
+msgstr "轉換到極座標和從極座標做轉換"
 
 #: ../../library/cmath.rst:44
 msgid ""
@@ -68,6 +81,8 @@ msgid ""
 "*Cartesian* coordinates.  It is completely determined by its *real part* ``z."
 "real`` and its *imaginary part* ``z.imag``.  In other words::"
 msgstr ""
+"Python 複數 ``z`` 是用 *直角坐標* 或 *笛卡爾坐標* 儲存在內部的。它完全是由其 "
+"*實部* ``z.real`` 和 *虛部* ``z.imag`` 所決定。換句話說： ::"
 
 #: ../../library/cmath.rst:51
 msgid ""
@@ -78,12 +93,15 @@ msgid ""
 "radians, from the positive x-axis to the line segment that joins the origin "
 "to *z*."
 msgstr ""
+"*極座標* 提供了另一種表示複數的方法。在極座標中，複數 *z* 由絕對值 (modulus) "
+"*r* 和相位角 (phase) *phi* 定義。絕對值 *r* 是從 *z* 到原點的距離，而相位角 "
+"*phi* 是從正 x 軸到連接原點到 *z* 的線段的逆時針角度（以弧度為單位）。"
 
 #: ../../library/cmath.rst:58
 msgid ""
 "The following functions can be used to convert from the native rectangular "
 "coordinates to polar coordinates and back."
-msgstr ""
+msgstr "以下的函式可用於原始直角座標與極座標之間的相互轉換。"
 
 #: ../../library/cmath.rst:63
 msgid ""
@@ -93,6 +111,10 @@ msgid ""
 "along the negative real axis.  The sign of the result is the same as the "
 "sign of ``x.imag``, even when ``x.imag`` is zero::"
 msgstr ""
+"以浮點數的形式回傳 *x* 的相位角（也稱為 *x* 的 *引數* ）。 ``phase(x)`` 等價"
+"於 ``math.atan2(x.imag, x.real)``。結果將位於 [-\\ *π*, *π*] 的範圍內，且此操"
+"作的分枝切割將位於負實軸上。結果的符號會與 ``x.imag`` 的符號相同，即使 ``x."
+"imag`` 為零： ::"
 
 #: ../../library/cmath.rst:77
 msgid ""
@@ -100,6 +122,8 @@ msgid ""
 "the built-in :func:`abs` function.  There is no separate :mod:`cmath` module "
 "function for this operation."
 msgstr ""
+"複數 *x* 的絕對值可以使用內建的 :func:`abs` 函式計算。沒有單獨的 :mod:"
+"`cmath` 模組函式適用於此操作。"
 
 #: ../../library/cmath.rst:84
 msgid ""
@@ -107,22 +131,26 @@ msgid ""
 "phi)`` where *r* is the modulus of *x* and phi is the phase of *x*.  "
 "``polar(x)`` is equivalent to ``(abs(x), phase(x))``."
 msgstr ""
+"回傳 *x* 在極座標中的表達方式。回傳一組數對 ``(r, phi)``， *r* 是 *x* 的絕對"
+"值， *phi* 是 *x* 的相位角。 ``polar(x)`` 相當於 ``(abs(x), phase(x))``。"
 
 #: ../../library/cmath.rst:92
 msgid ""
 "Return the complex number *x* with polar coordinates *r* and *phi*. "
 "Equivalent to ``r * (math.cos(phi) + math.sin(phi)*1j)``."
 msgstr ""
+"透過極座標 *r* 和 *phi* 回傳複數 *x*。相當於 ``r * (math.cos(phi) + math."
+"sin(phi)*1j)``。"
 
 #: ../../library/cmath.rst:97
 msgid "Power and logarithmic functions"
-msgstr ""
+msgstr "冪函數和對數函數"
 
 #: ../../library/cmath.rst:101
 msgid ""
 "Return *e* raised to the power *x*, where *e* is the base of natural "
 "logarithms."
-msgstr ""
+msgstr "回傳 *e* 的 *x* 次方，其中 *e* 是自然對數的底數。"
 
 #: ../../library/cmath.rst:107
 msgid ""
@@ -130,21 +158,23 @@ msgid ""
 "specified, returns the natural logarithm of *x*. There is one branch cut, "
 "from 0 along the negative real axis to -∞."
 msgstr ""
+"回傳 *x* 給定 *base* 的對數。如果未指定 *base* ，則傳回 *x* 的自然對數。存在"
+"一條分枝切割，從 0 沿負實數軸到 -∞。"
 
 #: ../../library/cmath.rst:114
 msgid ""
 "Return the base-10 logarithm of *x*. This has the same branch cut as :func:"
 "`log`."
-msgstr ""
+msgstr "回傳 *x* 以 10 為底的對數。它與 :func:`log` 具有相同的分枝切割。"
 
 #: ../../library/cmath.rst:120
 msgid ""
 "Return the square root of *x*. This has the same branch cut as :func:`log`."
-msgstr ""
+msgstr "回傳 *x* 的平方根。它與 :func:`log` 具有相同的分枝切割。"
 
 #: ../../library/cmath.rst:124
 msgid "Trigonometric functions"
-msgstr ""
+msgstr "三角函數"
 
 #: ../../library/cmath.rst:128
 msgid ""
@@ -152,11 +182,13 @@ msgid ""
 "from 1 along the real axis to ∞. The other extends left from -1 along the "
 "real axis to -∞."
 msgstr ""
+"回傳 *x* 的反餘弦值。存在兩條分枝切割：一條是從 1 沿著實數軸向右延伸到 ∞。另"
+"一條從 -1 沿實數軸向左延伸到 -∞。"
 
 #: ../../library/cmath.rst:135
 msgid ""
 "Return the arc sine of *x*. This has the same branch cuts as :func:`acos`."
-msgstr ""
+msgstr "回傳 *x* 的反正弦值。它與 :func:`acos` 具有相同的分枝切割。"
 
 #: ../../library/cmath.rst:140
 msgid ""
@@ -164,28 +196,31 @@ msgid ""
 "``1j`` along the imaginary axis to ``∞j``. The other extends from ``-1j`` "
 "along the imaginary axis to ``-∞j``."
 msgstr ""
+"回傳 *x* 的反正切值。有兩條分枝切割：一條是從 ``1j`` 沿著虛軸延伸到 ``∞j``。"
+"另一條從 ``-1j`` 沿著虛軸延伸到 ``-∞j``。"
 
 #: ../../library/cmath.rst:147
 msgid "Return the cosine of *x*."
-msgstr ""
+msgstr "回傳 *x* 的餘弦值。"
 
 #: ../../library/cmath.rst:152
 msgid "Return the sine of *x*."
-msgstr ""
+msgstr "回傳 *x* 的正弦值。"
 
 #: ../../library/cmath.rst:157
 msgid "Return the tangent of *x*."
-msgstr ""
+msgstr "回傳 *x* 的正切值。"
 
 #: ../../library/cmath.rst:161
 msgid "Hyperbolic functions"
-msgstr ""
+msgstr "雙曲函數"
 
 #: ../../library/cmath.rst:165
 msgid ""
 "Return the inverse hyperbolic cosine of *x*. There is one branch cut, "
 "extending left from 1 along the real axis to -∞."
 msgstr ""
+"回傳 *x* 的反雙曲餘弦值。存在一條分枝切割，從 1 沿實數軸向左延伸到 -∞。"
 
 #: ../../library/cmath.rst:171
 msgid ""
@@ -193,6 +228,8 @@ msgid ""
 "extends from ``1j`` along the imaginary axis to ``∞j``.  The other extends "
 "from ``-1j`` along the imaginary axis to ``-∞j``."
 msgstr ""
+"回傳 *x* 的反雙曲正弦值。存在兩條分枝切割：一條是從 ``1j`` 沿著虛軸延伸到 "
+"``∞j``。另一條從 ``-1j`` 沿著虛軸延伸到 ``-∞j``。"
 
 #: ../../library/cmath.rst:178
 msgid ""
@@ -200,52 +237,55 @@ msgid ""
 "extends from ``1`` along the real axis to ``∞``. The other extends from "
 "``-1`` along the real axis to ``-∞``."
 msgstr ""
+"回傳 *x* 的反雙曲正切值。存在兩條分枝切割：一條是從 ``1`` 沿著實數軸延伸到 "
+"``∞``。另一條從 ``-1`` 沿著實數軸延伸到 ``-∞``。"
 
 #: ../../library/cmath.rst:185
 msgid "Return the hyperbolic cosine of *x*."
-msgstr ""
+msgstr "回傳 *x* 的反雙曲餘弦值。"
 
 #: ../../library/cmath.rst:190
 msgid "Return the hyperbolic sine of *x*."
-msgstr ""
+msgstr "回傳 *x* 的反雙曲正弦值。"
 
 #: ../../library/cmath.rst:195
 msgid "Return the hyperbolic tangent of *x*."
-msgstr ""
+msgstr "回傳 *x* 的反雙曲正切值。"
 
 #: ../../library/cmath.rst:199
 msgid "Classification functions"
-msgstr ""
+msgstr "分類函式"
 
 #: ../../library/cmath.rst:203
 msgid ""
 "Return ``True`` if both the real and imaginary parts of *x* are finite, and "
 "``False`` otherwise."
-msgstr ""
+msgstr "如果 *x* 的實部和虛部都是有限的，則回傳 ``True``，否則回傳 ``False``。"
 
 #: ../../library/cmath.rst:211
 msgid ""
 "Return ``True`` if either the real or the imaginary part of *x* is an "
 "infinity, and ``False`` otherwise."
-msgstr ""
+msgstr "如果 *x* 的實部或虛部是無窮大，則回傳 ``True``，否則回傳 ``False``。"
 
 #: ../../library/cmath.rst:217
 msgid ""
 "Return ``True`` if either the real or the imaginary part of *x* is a NaN, "
 "and ``False`` otherwise."
-msgstr ""
+msgstr "如果 *x* 的實部或虛部為 NaN，則回傳 ``True``，否則回傳 ``False``。"
 
 #: ../../library/cmath.rst:223
 msgid ""
 "Return ``True`` if the values *a* and *b* are close to each other and "
 "``False`` otherwise."
-msgstr ""
+msgstr "如果 *a* 和 *b* 的值相互接近，則回傳 ``True``，否則回傳 ``False``。"
 
 #: ../../library/cmath.rst:226
 msgid ""
 "Whether or not two values are considered close is determined according to "
 "given absolute and relative tolerances."
 msgstr ""
+"兩個值是否被認為相互接近是由給定的絕對和相對容許偏差 (tolerance) 所決定的。"
 
 #: ../../library/cmath.rst:229
 msgid ""
@@ -255,18 +295,26 @@ msgid ""
 "tolerance is ``1e-09``, which assures that the two values are the same "
 "within about 9 decimal digits.  *rel_tol* must be greater than zero."
 msgstr ""
+"*rel_tol* 是相對容許偏差 -- 它是 *a* 和 *b* 之間的最大容許偏差值，相對於 *a* "
+"或 *b* 的較大絕對值。例如，要設定 5% 的容許偏差，請傳遞 ``rel_tol=0.05``。預"
+"設容許偏差是 ``1e-09``，它確保兩個值在大約 9 位十進制數字內相同。 *rel_tol* "
+"必須大於零。"
 
 #: ../../library/cmath.rst:235
 msgid ""
 "*abs_tol* is the minimum absolute tolerance -- useful for comparisons near "
 "zero. *abs_tol* must be at least zero."
 msgstr ""
+"*abs_tol* 是最小絕對容許偏差 -- 對於接近零的比較很有用。 *abs_tol* 必須至少為"
+"零。"
 
 #: ../../library/cmath.rst:238
 msgid ""
 "If no errors occur, the result will be: ``abs(a-b) <= max(rel_tol * "
 "max(abs(a), abs(b)), abs_tol)``."
 msgstr ""
+"如果未發生錯誤，結果將為： ``abs(a-b) <= max(rel_tol * max(abs(a), abs(b)), "
+"abs_tol)``。"
 
 #: ../../library/cmath.rst:241
 msgid ""
@@ -275,10 +323,13 @@ msgid ""
 "close to any other value, including ``NaN``.  ``inf`` and ``-inf`` are only "
 "considered close to themselves."
 msgstr ""
+"IEEE 754 特殊值 ``NaN`` 、 ``inf`` 和 ``-inf`` 將會根據 IEEE 規則進行處理。具"
+"體來說， ``NaN`` 不被認為接近任何其他值，包括 ``NaN``。 ``inf`` 和 ``-inf`` "
+"只被認為是接近它們自己的。"
 
 #: ../../library/cmath.rst:250
 msgid ":pep:`485` -- A function for testing approximate equality"
-msgstr ""
+msgstr ":pep:`485` -- 一個用來測試近似相等的函式"
 
 #: ../../library/cmath.rst:254
 msgid "Constants"
@@ -286,37 +337,37 @@ msgstr "常數"
 
 #: ../../library/cmath.rst:258
 msgid "The mathematical constant *π*, as a float."
-msgstr ""
+msgstr "數學常數 *π*，作為一個浮點數。"
 
 #: ../../library/cmath.rst:263
 msgid "The mathematical constant *e*, as a float."
-msgstr ""
+msgstr "數學常數 *e*，作為一個浮點數。"
 
 #: ../../library/cmath.rst:268
 msgid "The mathematical constant *τ*, as a float."
-msgstr ""
+msgstr "數學常數 *τ*，作為一個浮點數。"
 
 #: ../../library/cmath.rst:275
 msgid "Floating-point positive infinity. Equivalent to ``float('inf')``."
-msgstr ""
+msgstr "正無窮大的浮點數。相當於 ``float('inf')``。"
 
 #: ../../library/cmath.rst:282
 msgid ""
 "Complex number with zero real part and positive infinity imaginary part. "
 "Equivalent to ``complex(0.0, float('inf'))``."
-msgstr ""
+msgstr "實部為零和虛部為正無窮的複數。相當於 ``complex(0.0, float('inf'))``。"
 
 #: ../../library/cmath.rst:290
 msgid ""
 "A floating-point \"not a number\" (NaN) value.  Equivalent to "
 "``float('nan')``."
-msgstr ""
+msgstr "浮點「非數字」 (NaN) 值。相當於 ``float('nan')``。"
 
 #: ../../library/cmath.rst:298
 msgid ""
 "Complex number with zero real part and NaN imaginary part. Equivalent to "
 "``complex(0.0, float('nan'))``."
-msgstr ""
+msgstr "實部為零和虛部為 NaN 的複數。相當於 ``complex(0.0, float('nan'))``。"
 
 #: ../../library/cmath.rst:306
 msgid ""
@@ -329,6 +380,10 @@ msgid ""
 "as a real number (in which case the complex number has an imaginary part of "
 "zero)."
 msgstr ""
+"請注意，函式的選擇與模組 :mod:`math` 的類似，但並不完全相同。擁有兩個模組的原"
+"因是有些用戶對複數不感興趣，甚至根本就不知道它們是什麼。他們寧願讓 ``math."
+"sqrt(-1)`` 引發異常，也不願它回傳複數。另請注意， :mod:`cmath` 中所定義的函式"
+"始終都會回傳複數，即使答案可以表示為實數（在這種情況下，複數的虛部為零）。"
 
 #: ../../library/cmath.rst:314
 msgid ""
@@ -340,6 +395,10 @@ msgid ""
 "choice of branch cuts for numerical purposes, a good reference should be the "
 "following:"
 msgstr ""
+"關於分枝切割的註釋：它們是沿著給定的不連續函式的曲線。它們是許多複變函數的必"
+"要特徵。假設您需要使用複變函數進行計算，您將會了解分枝切割的概念。請參閱幾乎"
+"所有關於複變函數的（不是太初級的）書籍以獲得啟發。對於如何正確地基於數值目的"
+"選擇分枝切割的相關訊息，以下內容應該是一個很好的參考："
 
 #: ../../library/cmath.rst:324
 msgid ""

From 2d1c8fb93caef8c471590e9f88824698a354fdb9 Mon Sep 17 00:00:00 2001
From: Enkai Huang <enkaihuang2021@gmail.com>
Date: Thu, 14 Mar 2024 03:14:55 +0000
Subject: [PATCH 4/4] fix typo

---
 library/cmath.po | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/library/cmath.po b/library/cmath.po
index 219d52293f..1d0989d677 100644
--- a/library/cmath.po
+++ b/library/cmath.po
@@ -9,7 +9,7 @@ msgstr ""
 "Report-Msgid-Bugs-To: \n"
 "POT-Creation-Date: 2023-06-30 15:31+0000\n"
 "PO-Revision-Date: 2024-03-14 09:26+0800\n"
-"Last-Translator: Adrian Liaw <adrianliaw2000@gmail.com>\n"
+"Last-Translator: Enkai Huang <enkaihuang2021@gmail.com>\n"
 "Language-Team: Chinese - TAIWAN (https://github.com/python/python-docs-zh-"
 "tw)\n"
 "Language: zh_TW\n"
@@ -158,7 +158,7 @@ msgid ""
 "specified, returns the natural logarithm of *x*. There is one branch cut, "
 "from 0 along the negative real axis to -∞."
 msgstr ""
-"回傳 *x* 給定 *base* 的對數。如果未指定 *base* ，則傳回 *x* 的自然對數。存在"
+"回傳 *x* 給定 *base* 的對數。如果未指定 *base*，則傳回 *x* 的自然對數。存在"
 "一條分枝切割，從 0 沿負實數軸到 -∞。"
 
 #: ../../library/cmath.rst:114