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                ### 導航 - [索引](../genindex.xhtml "總目錄") - [模塊](../py-modindex.xhtml "Python 模塊索引") | - [下一頁](fractions.xhtml "fractions --- 分數") | - [上一頁](cmath.xhtml "cmath --- Mathematical functions for complex numbers") | - ![](https://box.kancloud.cn/a721fc7ec672275e257bbbfde49a4d4e_16x16.png) - [Python](https://www.python.org/) ? - zh\_CN 3.7.3 [文檔](../index.xhtml) ? - [Python 標準庫](index.xhtml) ? - [數字和數學模塊](numeric.xhtml) ? - $('.inline-search').show(0); | # [`decimal`](#module-decimal "decimal: Implementation of the General Decimal Arithmetic Specification.") --- 十進制定點和浮點運算 **源碼:** [Lib/decimal.py](https://github.com/python/cpython/tree/3.7/Lib/decimal.py) \[https://github.com/python/cpython/tree/3.7/Lib/decimal.py\] - - - - - - [`decimal`](#module-decimal "decimal: Implementation of the General Decimal Arithmetic Specification.") 模塊為快速正確舍入的十進制浮點運算提供支持。 它提供了 [`float`](functions.xhtml#float "float") 數據類型以外的幾個優點: - Decimal “基于一個浮點模型,它是為人們設計的,并且必然具有最重要的指導原則 —— 計算機必須提供與人們在學校學習的算法相同的算法。” —— 摘自十進制算術規范。 - 十進制數字可以準確表示。 相比之下,數字如 `1.1` 和 `2.2` 在二進制浮點中沒有精確的表示。 最終用戶通常不希望``1.1 + 2.2``顯示為 `3.3000000000000003` ,就像二進制浮點一樣。 - 精確性延續到算術中。 在十進制浮點數中,`0.1 + 0.1 + 0.1 - 0.3` 恰好等于零。 在二進制浮點數中,結果為 `5.5511151231257827e-017` 。 雖然接近于零,但差異妨礙了可靠的相等性檢驗,并且差異可能會累積。 因此,在具有嚴格相等不變量的會計應用程序中, decimal 是首選。 - 十進制模塊包含一個重要位置的概念,因此 `1.30 + 1.20` 是 `2.50` 。 保留尾隨零以表示重要性。 這是貨幣申請的慣常陳述。 對于乘法,“教科書”方法使用被乘數中的所有數字。 例如, `1.3 * 1.2` 給出 `1.56` 而 `1.30 * 1.20` 給出 `1.5600` 。 - 與基于硬件的二進制浮點不同,十進制模塊具有用戶可更改的精度(默認為28個位置),可以與給定問題所需的一樣大: ``` >>> from decimal import * >>> getcontext().prec = 6 >>> Decimal(1) / Decimal(7) Decimal('0.142857') >>> getcontext().prec = 28 >>> Decimal(1) / Decimal(7) Decimal('0.1428571428571428571428571429') ``` - 二進制和十進制浮點都是根據已發布的標準實現的。 雖然內置浮點類型只公開其功能的一小部分,但十進制模塊公開了標準的所有必需部分。 在需要時,程序員可以完全控制舍入和信號處理。 這包括通過使用異常來阻止任何不精確操作來強制執行精確算術的選項。 - 十進制模塊旨在支持“無偏見,精確的非連續十進制算術(有時稱為定點算術)和舍入浮點算術”。 —— 摘自十進制算術規范。 模塊設計以三個概念為中心:十進制數,算術上下文和信號。 十進制數是不可變的。 它有一個符號,系數數字和一個指數。 為了保持重要性,系數數字不會截斷尾隨零。十進制數也包括特殊值,例如 `Infinity` ,`-Infinity` ,和 `NaN` 。 該標準還區分 `-0` 和 `+0` 。 算術的上下文是指定精度、舍入規則、指數限制、指示操作結果的標志以及確定符號是否被視為異常的陷阱啟用器的環境。 舍入選項包括 [`ROUND_CEILING`](#decimal.ROUND_CEILING "decimal.ROUND_CEILING") 、 [`ROUND_DOWN`](#decimal.ROUND_DOWN "decimal.ROUND_DOWN") 、 [`ROUND_FLOOR`](#decimal.ROUND_FLOOR "decimal.ROUND_FLOOR") 、 [`ROUND_HALF_DOWN`](#decimal.ROUND_HALF_DOWN "decimal.ROUND_HALF_DOWN"), [`ROUND_HALF_EVEN`](#decimal.ROUND_HALF_EVEN "decimal.ROUND_HALF_EVEN") 、 [`ROUND_HALF_UP`](#decimal.ROUND_HALF_UP "decimal.ROUND_HALF_UP") 、 [`ROUND_UP`](#decimal.ROUND_UP "decimal.ROUND_UP") 以及 [`ROUND_05UP`](#decimal.ROUND_05UP "decimal.ROUND_05UP"). 信號是在計算過程中出現的異常條件組。 根據應用程序的需要,信號可能會被忽略,被視為信息,或被視為異常。 十進制模塊中的信號有:[`Clamped`](#decimal.Clamped "decimal.Clamped") 、 [`InvalidOperation`](#decimal.InvalidOperation "decimal.InvalidOperation") 、 [`DivisionByZero`](#decimal.DivisionByZero "decimal.DivisionByZero") 、 [`Inexact`](#decimal.Inexact "decimal.Inexact") 、 [`Rounded`](#decimal.Rounded "decimal.Rounded") 、 [`Subnormal`](#decimal.Subnormal "decimal.Subnormal") 、 [`Overflow`](#decimal.Overflow "decimal.Overflow") 、 [`Underflow`](#decimal.Underflow "decimal.Underflow") 以及 [`FloatOperation`](#decimal.FloatOperation "decimal.FloatOperation") 。 對于每個信號,都有一個標志和一個陷阱啟動器。 遇到信號時,其標志設置為 1 ,然后,如果陷阱啟用器設置為 1 ,則引發異常。 標志是粘性的,因此用戶需要在監控計算之前重置它們。 參見 - IBM的通用十進制算術規范, [The General Decimal Arithmetic Specification](http://speleotrove.com/decimal/decarith.html) \[http://speleotrove.com/decimal/decarith.html\]. ## 快速入門教程 通常使用小數的開始是導入模塊,使用 [`getcontext()`](#decimal.getcontext "decimal.getcontext") 查看當前上下文,并在必要時為精度、舍入或啟用的陷阱設置新值: ``` >>> from decimal import * >>> getcontext() Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=-999999, Emax=999999, capitals=1, clamp=0, flags=[], traps=[Overflow, DivisionByZero, InvalidOperation]) >>> getcontext().prec = 7 # Set a new precision ``` 可以從整數、字符串、浮點數或元組構造十進制實例。 從整數或浮點構造將執行該整數或浮點值的精確轉換。 十進制數包括特殊值,例如 `NaN` 代表“非數字”,正的和負的 `Infinity`,和 `-0` ``` >>> getcontext().prec = 28 >>> Decimal(10) Decimal('10') >>> Decimal('3.14') Decimal('3.14') >>> Decimal(3.14) Decimal('3.140000000000000124344978758017532527446746826171875') >>> Decimal((0, (3, 1, 4), -2)) Decimal('3.14') >>> Decimal(str(2.0 ** 0.5)) Decimal('1.4142135623730951') >>> Decimal(2) ** Decimal('0.5') Decimal('1.414213562373095048801688724') >>> Decimal('NaN') Decimal('NaN') >>> Decimal('-Infinity') Decimal('-Infinity') ``` 如果 [`FloatOperation`](#decimal.FloatOperation "decimal.FloatOperation") 信號被捕獲,構造函數中的小數和浮點數的意外混合或排序比較會引發異常 ``` >>> c = getcontext() >>> c.traps[FloatOperation] = True >>> Decimal(3.14) Traceback (most recent call last): File "<stdin>", line 1, in <module> decimal.FloatOperation: [<class 'decimal.FloatOperation'>] >>> Decimal('3.5') < 3.7 Traceback (most recent call last): File "<stdin>", line 1, in <module> decimal.FloatOperation: [<class 'decimal.FloatOperation'>] >>> Decimal('3.5') == 3.5 True ``` 3\.3 新版功能. 新 Decimal 的重要性僅由輸入的位數決定。 上下文精度和舍入僅在算術運算期間發揮作用。 ``` >>> getcontext().prec = 6 >>> Decimal('3.0') Decimal('3.0') >>> Decimal('3.1415926535') Decimal('3.1415926535') >>> Decimal('3.1415926535') + Decimal('2.7182818285') Decimal('5.85987') >>> getcontext().rounding = ROUND_UP >>> Decimal('3.1415926535') + Decimal('2.7182818285') Decimal('5.85988') ``` 如果超出了C版本的內部限制,則構造一個十進制將引發 [`InvalidOperation`](#decimal.InvalidOperation "decimal.InvalidOperation") ``` >>> Decimal("1e9999999999999999999") Traceback (most recent call last): File "<stdin>", line 1, in <module> decimal.InvalidOperation: [<class 'decimal.InvalidOperation'>] ``` 在 3.3 版更改. 小數與 Python 的其余部分很好地交互。 這是一個小的十進制浮點飛行雜技團: ``` >>> data = list(map(Decimal, '1.34 1.87 3.45 2.35 1.00 0.03 9.25'.split())) >>> max(data) Decimal('9.25') >>> min(data) Decimal('0.03') >>> sorted(data) [Decimal('0.03'), Decimal('1.00'), Decimal('1.34'), Decimal('1.87'), Decimal('2.35'), Decimal('3.45'), Decimal('9.25')] >>> sum(data) Decimal('19.29') >>> a,b,c = data[:3] >>> str(a) '1.34' >>> float(a) 1.34 >>> round(a, 1) Decimal('1.3') >>> int(a) 1 >>> a * 5 Decimal('6.70') >>> a * b Decimal('2.5058') >>> c % a Decimal('0.77') ``` Decimal 也可以使用一些數學函數: ``` >>> getcontext().prec = 28 >>> Decimal(2).sqrt() Decimal('1.414213562373095048801688724') >>> Decimal(1).exp() Decimal('2.718281828459045235360287471') >>> Decimal('10').ln() Decimal('2.302585092994045684017991455') >>> Decimal('10').log10() Decimal('1') ``` `quantize()` 方法將數字四舍五入為固定指數。 此方法對于將結果舍入到固定的位置的貨幣應用程序非常有用: ``` >>> Decimal('7.325').quantize(Decimal('.01'), rounding=ROUND_DOWN) Decimal('7.32') >>> Decimal('7.325').quantize(Decimal('1.'), rounding=ROUND_UP) Decimal('8') ``` 如上所示,[`getcontext()`](#decimal.getcontext "decimal.getcontext") 函數訪問當前上下文并允許更改設置。 這種方法滿足大多數應用程序的需求。 對于更高級的工作,使用 Context() 構造函數創建備用上下文可能很有用。 要使用備用活動,請使用 [`setcontext()`](#decimal.setcontext "decimal.setcontext") 函數。 根據標準,[`decimal`](#module-decimal "decimal: Implementation of the General Decimal Arithmetic Specification.") 模塊提供了兩個現成的標準上下文 [`BasicContext`](#decimal.BasicContext "decimal.BasicContext") 和 [`ExtendedContext`](#decimal.ExtendedContext "decimal.ExtendedContext") 。 前者對調試特別有用,因為許多陷阱都已啟用: ``` >>> myothercontext = Context(prec=60, rounding=ROUND_HALF_DOWN) >>> setcontext(myothercontext) >>> Decimal(1) / Decimal(7) Decimal('0.142857142857142857142857142857142857142857142857142857142857') >>> ExtendedContext Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999, Emax=999999, capitals=1, clamp=0, flags=[], traps=[]) >>> setcontext(ExtendedContext) >>> Decimal(1) / Decimal(7) Decimal('0.142857143') >>> Decimal(42) / Decimal(0) Decimal('Infinity') >>> setcontext(BasicContext) >>> Decimal(42) / Decimal(0) Traceback (most recent call last): File "<pyshell#143>", line 1, in -toplevel- Decimal(42) / Decimal(0) DivisionByZero: x / 0 ``` 上下文還具有用于監視計算期間遇到的異常情況的信號標志。 標志保持設置直到明確清除,因此最好通過使用 `clear_flags()` 方法清除每組受監控計算之前的標志。: ``` >>> setcontext(ExtendedContext) >>> getcontext().clear_flags() >>> Decimal(355) / Decimal(113) Decimal('3.14159292') >>> getcontext() Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999, Emax=999999, capitals=1, clamp=0, flags=[Inexact, Rounded], traps=[]) ``` *flags* 條目顯示對 `Pi` 的有理逼近被舍入(超出上下文精度的數字被拋棄)并且結果是不精確的(一些丟棄的數字不為零)。 使用上下文的 `traps` 字段中的字典設置單個陷阱: ``` >>> setcontext(ExtendedContext) >>> Decimal(1) / Decimal(0) Decimal('Infinity') >>> getcontext().traps[DivisionByZero] = 1 >>> Decimal(1) / Decimal(0) Traceback (most recent call last): File "<pyshell#112>", line 1, in -toplevel- Decimal(1) / Decimal(0) DivisionByZero: x / 0 ``` 大多數程序僅在程序開始時調整當前上下文一次。 并且,在許多應用程序中,數據在循環內單個強制轉換為 [`Decimal`](#decimal.Decimal "decimal.Decimal") 。 通過創建上下文集和小數,程序的大部分操作數據與其他 Python 數字類型沒有區別。 ## Decimal 對象 *class* `decimal.``Decimal`(*value="0"*, *context=None*)根據 *value* 構造一個新的 [`Decimal`](#decimal.Decimal "decimal.Decimal") 對象。 *value* 可以是整數,字符串,元組,[`float`](functions.xhtml#float "float") ,或另一個 [`Decimal`](#decimal.Decimal "decimal.Decimal") 對象。 如果沒有給出 *value*,則返回 `Decimal('0')`。 如果 *value* 是一個字符串,它應該在前導和尾隨空格字符以及下劃線被刪除之后符合十進制數字字符串語法: ``` sign ::= '+' | '-' digit ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9' indicator ::= 'e' | 'E' digits ::= digit [digit]... decimal-part ::= digits '.' [digits] | ['.'] digits exponent-part ::= indicator [sign] digits infinity ::= 'Infinity' | 'Inf' nan ::= 'NaN' [digits] | 'sNaN' [digits] numeric-value ::= decimal-part [exponent-part] | infinity numeric-string ::= [sign] numeric-value | [sign] nan ``` 當上面出現 `digit` 時也允許其他十進制數碼。 其中包括來自各種其他語言系統的十進制數碼(例如阿拉伯-印地語和天城文的數碼)以及全寬數碼 `'\uff10'` 到 `'\uff19'`。 如果 *value* 是一個 [`tuple`](stdtypes.xhtml#tuple "tuple") ,它應該有三個組件,一個符號( `0` 表示正數或 `1` 表示負數),一個數字的 [`tuple`](stdtypes.xhtml#tuple "tuple") 和整數指數。 例如, `Decimal((0, (1, 4, 1, 4), -3))` 返回 `Decimal('1.414')`。 如果 *value* 是 [`float`](functions.xhtml#float "float") ,則二進制浮點值無損地轉換為其精確的十進制等效值。 此轉換通常需要53位或更多位數的精度。 例如, `Decimal(float('1.1'))` 轉換為``Decimal('1.100000000000000088817841970012523233890533447265625')``。 *context* 精度不會影響存儲的位數。 這完全由 *value* 中的位數決定。 例如,`Decimal('3.00000')` 記錄所有五個零,即使上下文精度只有三。 *context* 參數的目的是確定 *value* 是格式錯誤的字符串時該怎么做。 如果上下文陷阱 [`InvalidOperation`](#decimal.InvalidOperation "decimal.InvalidOperation"),則引發異常;否則,構造函數返回一個新的 Decimal,其值為 `NaN`。 構造完成后, [`Decimal`](#decimal.Decimal "decimal.Decimal") 對象是不可變的。 在 3.2 版更改: 現在允許構造函數的參數為 [`float`](functions.xhtml#float "float") 實例。 在 3.3 版更改: [`float`](functions.xhtml#float "float") 參數在設置 [`FloatOperation`](#decimal.FloatOperation "decimal.FloatOperation") 陷阱時引發異常。 默認情況下,陷阱已關閉。 在 3.6 版更改: 允許下劃線進行分組,就像代碼中的整數和浮點文字一樣。 十進制浮點對象與其他內置數值類型共享許多屬性,例如 [`float`](functions.xhtml#float "float") 和 [`int`](functions.xhtml#int "int") 。 所有常用的數學運算和特殊方法都適用。 同樣,十進制對象可以復制、pickle、打印、用作字典鍵、用作集合元素、比較、排序和強制轉換為另一種類型(例如 [`float`](functions.xhtml#float "float") 或 [`int`](functions.xhtml#int "int") )。 算術對十進制對象和算術對整數和浮點數有一些小的差別。 當余數運算符 `%` 應用于Decimal對象時,結果的符號是 *被除數* 的符號,而不是除數的符號: ``` >>> (-7) % 4 1 >>> Decimal(-7) % Decimal(4) Decimal('-3') ``` 整數除法運算符 `//` 的行為類似,返回真商的整數部分(截斷為零)而不是它的向下取整,以便保留通常的標識 `x == (x // y) * y + x % y`: ``` >>> -7 // 4 -2 >>> Decimal(-7) // Decimal(4) Decimal('-1') ``` `%` 和 `//` 運算符實現了 `remainder` 和 `divide-integer` 操作(分別),如規范中所述。 十進制對象通常不能與浮點數或 [`fractions.Fraction`](fractions.xhtml#fractions.Fraction "fractions.Fraction") 實例在算術運算中結合使用:例如,嘗試將 [`Decimal`](#decimal.Decimal "decimal.Decimal") 加到 [`float`](functions.xhtml#float "float") ,將引發 [`TypeError`](exceptions.xhtml#TypeError "TypeError")。 但是,可以使用 Python 的比較運算符來比較 [`Decimal`](#decimal.Decimal "decimal.Decimal") 實例 `x` 和另一個數字 `y` 。 這樣可以避免在對不同類型的數字進行相等比較時混淆結果。 在 3.2 版更改: 現在完全支持 [`Decimal`](#decimal.Decimal "decimal.Decimal") 實例和其他數字類型之間的混合類型比較。 除了標準的數字屬性,十進制浮點對象還有許多專門的方法: `adjusted`()在移出系數最右邊的數字之后返回調整后的指數,直到只剩下前導數字:`Decimal('321e+5').adjusted()` 返回 7 。 用于確定最高有效位相對于小數點的位置。 `as_integer_ratio`()返回一對 `(n, d)` 整數,表示給定的 [`Decimal`](#decimal.Decimal "decimal.Decimal") 實例作為分數、最簡形式項并帶有正分母: ``` >>> Decimal('-3.14').as_integer_ratio() (-157, 50) ``` 轉換是精確的。 在 Infinity 上引發 OverflowError ,在 NaN 上引起 ValueError 。 3\.6 新版功能. `as_tuple`()返回一個 [named tuple](../glossary.xhtml#term-named-tuple) 表示的數字: `DecimalTuple(sign, digits, exponent)`。 `canonical`()返回參數的規范編碼。 目前,一個 [`Decimal`](#decimal.Decimal "decimal.Decimal") 實例的編碼始終是規范的,因此該操作返回其參數不變。 `compare`(*other*, *context=None*)比較兩個 Decimal 實例的值。 [`compare()`](#decimal.Decimal.compare "decimal.Decimal.compare") 返回一個 Decimal 實例,如果任一操作數是 NaN ,那么結果是 NaN ``` a or b is a NaN ==> Decimal('NaN') a < b ==> Decimal('-1') a == b ==> Decimal('0') a > b ==> Decimal('1') ``` `compare_signal`(*other*, *context=None*)除了所有 NaN 信號之外,此操作與 [`compare()`](#decimal.Decimal.compare "decimal.Decimal.compare") 方法相同。 也就是說,如果兩個操作數都不是信令NaN,那么任何靜默的 NaN 操作數都被視為信令NaN。 `compare_total`(*other*, *context=None*)使用它們的抽象表示而不是它們的數值來比較兩個操作數。 類似于 [`compare()`](#decimal.Decimal.compare "decimal.Decimal.compare") 方法,但結果給出了一個總排序 [`Decimal`](#decimal.Decimal "decimal.Decimal") 實例。 兩個 [`Decimal`](#decimal.Decimal "decimal.Decimal") 實例具有相同的數值但不同的表示形式在此排序中比較不相等: ``` >>> Decimal('12.0').compare_total(Decimal('12')) Decimal('-1') ``` 靜默和發出信號的 NaN 也包括在總排序中。 這個函數的結果是 `Decimal('0')` 如果兩個操作數具有相同的表示,或是 `Decimal('-1')` 如果第一個操作數的總順序低于第二個操作數,或是 `Decimal('1')` 如果第一個操作數在總順序中高于第二個操作數。 有關總排序的詳細信息,請參閱規范。 此操作不受上下文影響且靜默:不更改任何標志且不執行舍入。 作為例外,如果無法準確轉換第二個操作數,則C版本可能會引發InvalidOperation。 `compare_total_mag`(*other*, *context=None*)比較兩個操作數使用它們的抽象表示而不是它們的值,如 [`compare_total()`](#decimal.Decimal.compare_total "decimal.Decimal.compare_total"),但忽略每個操作數的符號。 `x.compare_total_mag(y)` 相當于 `x.copy_abs().compare_total(y.copy_abs())`。 此操作不受上下文影響且靜默:不更改任何標志且不執行舍入。 作為例外,如果無法準確轉換第二個操作數,則C版本可能會引發InvalidOperation。 `conjugate`()只返回self,這種方法只符合 Decimal 規范。 `copy_abs`()返回參數的絕對值。 此操作不受上下文影響并且是靜默的:沒有更改標志且不執行舍入。 `copy_negate`()回到參數的否定。 此操作不受上下文影響并且是靜默的:沒有標志更改且不執行舍入。 `copy_sign`(*other*, *context=None*)返回第一個操作數的副本,其符號設置為與第二個操作數的符號相同。 例如: ``` >>> Decimal('2.3').copy_sign(Decimal('-1.5')) Decimal('-2.3') ``` 此操作不受上下文影響且靜默:不更改任何標志且不執行舍入。 作為例外,如果無法準確轉換第二個操作數,則C版本可能會引發InvalidOperation。 `exp`(*context=None*)返回給定數字的(自然)指數函數``e\*\*x``的值。結果使用 [`ROUND_HALF_EVEN`](#decimal.ROUND_HALF_EVEN "decimal.ROUND_HALF_EVEN") 舍入模式正確舍入。 ``` >>> Decimal(1).exp() Decimal('2.718281828459045235360287471') >>> Decimal(321).exp() Decimal('2.561702493119680037517373933E+139') ``` `from_float`(*f*)將浮點數轉換為十進制數的類方法。 注意, Decimal.from\_float(0.1) 與 Decimal('0.1') 不同。 由于 0.1 在二進制浮點中不能精確表示,因此該值存儲為最接近的可表示值,即 0x1.999999999999ap-4 。 十進制的等效值是`0.1000000000000000055511151231257827021181583404541015625`。 注解 從 Python 3.2 開始,[`Decimal`](#decimal.Decimal "decimal.Decimal") 實例也可以直接從 [`float`](functions.xhtml#float "float") 構造。 ``` >>> Decimal.from_float(0.1) Decimal('0.1000000000000000055511151231257827021181583404541015625') >>> Decimal.from_float(float('nan')) Decimal('NaN') >>> Decimal.from_float(float('inf')) Decimal('Infinity') >>> Decimal.from_float(float('-inf')) Decimal('-Infinity') ``` 3\.1 新版功能. `fma`(*other*, *third*, *context=None*)混合乘法加法。 返回 self\*other+third ,中間乘積 self\*other 沒有四舍五入。 ``` >>> Decimal(2).fma(3, 5) Decimal('11') ``` `is_canonical`()如果參數是規范的,則為返回 [`True`](constants.xhtml#True "True"),否則為 [`False`](constants.xhtml#False "False") 。 目前,[`Decimal`](#decimal.Decimal "decimal.Decimal") 實例總是規范的,所以這個操作總是返回 [`True`](constants.xhtml#True "True") 。 `is_finite`()如果參數是一個有限的數,則返回為 [`True`](constants.xhtml#True "True") ;如果參數為無窮大或 NaN ,則返回為 [`False`](constants.xhtml#False "False")。 `is_infinite`()如果參數為正負無窮大,則返回為 [`True`](constants.xhtml#True "True") ,否則為 [`False`](constants.xhtml#False "False") 。 `is_nan`()如果參數為 NaN (無論是否靜默),則返回為 [`True`](constants.xhtml#True "True") ,否則為 [`False`](constants.xhtml#False "False") 。 `is_normal`(*context=None*)如果參數是一個有限正規數,返回 [`True`](constants.xhtml#True "True"),如果參數是0、次正規數、無窮大或是NaN,返回 [`False`](constants.xhtml#False "False")。 `is_qnan`()如果參數為靜默 NaN,返回 [`True`](constants.xhtml#True "True"),否則返回 [`False`](constants.xhtml#False "False")。 `is_signed`()如果參數帶有負號,則返回為 [`True`](constants.xhtml#True "True"),否則返回 [`False`](constants.xhtml#False "False")。注意,0 和 NaN 都可帶有符號。 `is_snan`()如果參數為顯式 NaN,則返回 [`True`](constants.xhtml#True "True"),否則返回 [`False`](constants.xhtml#False "False")。 `is_subnormal`(*context=None*)如果參數為次正規數,則返回 [`True`](constants.xhtml#True "True"),否則返回 [`False`](constants.xhtml#False "False")。 `is_zero`()如果參數是0(正負皆可),則返回 [`True`](constants.xhtml#True "True"),否則返回 [`False`](constants.xhtml#False "False")。 `ln`(*context=None*)Return the natural (base e) logarithm of the operand. The result is correctly rounded using the [`ROUND_HALF_EVEN`](#decimal.ROUND_HALF_EVEN "decimal.ROUND_HALF_EVEN") rounding mode. `log10`(*context=None*)Return the base ten logarithm of the operand. The result is correctly rounded using the [`ROUND_HALF_EVEN`](#decimal.ROUND_HALF_EVEN "decimal.ROUND_HALF_EVEN") rounding mode. `logb`(*context=None*)For a nonzero number, return the adjusted exponent of its operand as a [`Decimal`](#decimal.Decimal "decimal.Decimal") instance. If the operand is a zero then `Decimal('-Infinity')` is returned and the [`DivisionByZero`](#decimal.DivisionByZero "decimal.DivisionByZero") flag is raised. If the operand is an infinity then `Decimal('Infinity')` is returned. `logical_and`(*other*, *context=None*)[`logical_and()`](#decimal.Decimal.logical_and "decimal.Decimal.logical_and") is a logical operation which takes two *logical operands* (see [Logical operands](#logical-operands-label)). The result is the digit-wise `and` of the two operands. `logical_invert`(*context=None*)[`logical_invert()`](#decimal.Decimal.logical_invert "decimal.Decimal.logical_invert") is a logical operation. The result is the digit-wise inversion of the operand. `logical_or`(*other*, *context=None*)[`logical_or()`](#decimal.Decimal.logical_or "decimal.Decimal.logical_or") is a logical operation which takes two *logical operands* (see [Logical operands](#logical-operands-label)). The result is the digit-wise `or` of the two operands. `logical_xor`(*other*, *context=None*)[`logical_xor()`](#decimal.Decimal.logical_xor "decimal.Decimal.logical_xor") is a logical operation which takes two *logical operands* (see [Logical operands](#logical-operands-label)). The result is the digit-wise exclusive or of the two operands. `max`(*other*, *context=None*)Like `max(self, other)` except that the context rounding rule is applied before returning and that `NaN` values are either signaled or ignored (depending on the context and whether they are signaling or quiet). `max_mag`(*other*, *context=None*)Similar to the [`max()`](#decimal.Decimal.max "decimal.Decimal.max") method, but the comparison is done using the absolute values of the operands. `min`(*other*, *context=None*)Like `min(self, other)` except that the context rounding rule is applied before returning and that `NaN` values are either signaled or ignored (depending on the context and whether they are signaling or quiet). `min_mag`(*other*, *context=None*)Similar to the [`min()`](#decimal.Decimal.min "decimal.Decimal.min") method, but the comparison is done using the absolute values of the operands. `next_minus`(*context=None*)Return the largest number representable in the given context (or in the current thread's context if no context is given) that is smaller than the given operand. `next_plus`(*context=None*)Return the smallest number representable in the given context (or in the current thread's context if no context is given) that is larger than the given operand. `next_toward`(*other*, *context=None*)If the two operands are unequal, return the number closest to the first operand in the direction of the second operand. If both operands are numerically equal, return a copy of the first operand with the sign set to be the same as the sign of the second operand. `normalize`(*context=None*)Normalize the number by stripping the rightmost trailing zeros and converting any result equal to `Decimal('0')` to `Decimal('0e0')`. Used for producing canonical values for attributes of an equivalence class. For example, `Decimal('32.100')` and `Decimal('0.321000e+2')` both normalize to the equivalent value `Decimal('32.1')`. `number_class`(*context=None*)Return a string describing the *class* of the operand. The returned value is one of the following ten strings. - `"-Infinity"`, indicating that the operand is negative infinity. - `"-Normal"`, indicating that the operand is a negative normal number. - `"-Subnormal"`, indicating that the operand is negative and subnormal. - `"-Zero"`, indicating that the operand is a negative zero. - `"+Zero"`, indicating that the operand is a positive zero. - `"+Subnormal"`, indicating that the operand is positive and subnormal. - `"+Normal"`, indicating that the operand is a positive normal number. - `"+Infinity"`, indicating that the operand is positive infinity. - `"NaN"`, indicating that the operand is a quiet NaN (Not a Number). - `"sNaN"`, indicating that the operand is a signaling NaN. `quantize`(*exp*, *rounding=None*, *context=None*)Return a value equal to the first operand after rounding and having the exponent of the second operand. ``` >>> Decimal('1.41421356').quantize(Decimal('1.000')) Decimal('1.414') ``` Unlike other operations, if the length of the coefficient after the quantize operation would be greater than precision, then an [`InvalidOperation`](#decimal.InvalidOperation "decimal.InvalidOperation") is signaled. This guarantees that, unless there is an error condition, the quantized exponent is always equal to that of the right-hand operand. Also unlike other operations, quantize never signals Underflow, even if the result is subnormal and inexact. If the exponent of the second operand is larger than that of the first then rounding may be necessary. In this case, the rounding mode is determined by the `rounding` argument if given, else by the given `context` argument; if neither argument is given the rounding mode of the current thread's context is used. An error is returned whenever the resulting exponent is greater than `Emax` or less than `Etiny`. `radix`()Return `Decimal(10)`, the radix (base) in which the [`Decimal`](#decimal.Decimal "decimal.Decimal")class does all its arithmetic. Included for compatibility with the specification. `remainder_near`(*other*, *context=None*)Return the remainder from dividing *self* by *other*. This differs from `self % other` in that the sign of the remainder is chosen so as to minimize its absolute value. More precisely, the return value is `self - n * other` where `n` is the integer nearest to the exact value of `self / other`, and if two integers are equally near then the even one is chosen. If the result is zero then its sign will be the sign of *self*. ``` >>> Decimal(18).remainder_near(Decimal(10)) Decimal('-2') >>> Decimal(25).remainder_near(Decimal(10)) Decimal('5') >>> Decimal(35).remainder_near(Decimal(10)) Decimal('-5') ``` `rotate`(*other*, *context=None*)Return the result of rotating the digits of the first operand by an amount specified by the second operand. The second operand must be an integer in the range -precision through precision. The absolute value of the second operand gives the number of places to rotate. If the second operand is positive then rotation is to the left; otherwise rotation is to the right. The coefficient of the first operand is padded on the left with zeros to length precision if necessary. The sign and exponent of the first operand are unchanged. `same_quantum`(*other*, *context=None*)Test whether self and other have the same exponent or whether both are `NaN`. 此操作不受上下文影響且靜默:不更改任何標志且不執行舍入。 作為例外,如果無法準確轉換第二個操作數,則C版本可能會引發InvalidOperation。 `scaleb`(*other*, *context=None*)Return the first operand with exponent adjusted by the second. Equivalently, return the first operand multiplied by `10**other`. The second operand must be an integer. `shift`(*other*, *context=None*)Return the result of shifting the digits of the first operand by an amount specified by the second operand. The second operand must be an integer in the range -precision through precision. The absolute value of the second operand gives the number of places to shift. If the second operand is positive then the shift is to the left; otherwise the shift is to the right. Digits shifted into the coefficient are zeros. The sign and exponent of the first operand are unchanged. `sqrt`(*context=None*)Return the square root of the argument to full precision. `to_eng_string`(*context=None*)Convert to a string, using engineering notation if an exponent is needed. Engineering notation has an exponent which is a multiple of 3. This can leave up to 3 digits to the left of the decimal place and may require the addition of either one or two trailing zeros. For example, this converts `Decimal('123E+1')` to `Decimal('1.23E+3')`. `to_integral`(*rounding=None*, *context=None*)Identical to the [`to_integral_value()`](#decimal.Decimal.to_integral_value "decimal.Decimal.to_integral_value") method. The `to_integral`name has been kept for compatibility with older versions. `to_integral_exact`(*rounding=None*, *context=None*)Round to the nearest integer, signaling [`Inexact`](#decimal.Inexact "decimal.Inexact") or [`Rounded`](#decimal.Rounded "decimal.Rounded") as appropriate if rounding occurs. The rounding mode is determined by the `rounding` parameter if given, else by the given `context`. If neither parameter is given then the rounding mode of the current context is used. `to_integral_value`(*rounding=None*, *context=None*)Round to the nearest integer without signaling [`Inexact`](#decimal.Inexact "decimal.Inexact") or [`Rounded`](#decimal.Rounded "decimal.Rounded"). If given, applies *rounding*; otherwise, uses the rounding method in either the supplied *context* or the current context. ### Logical operands The `logical_and()`, `logical_invert()`, `logical_or()`, and `logical_xor()` methods expect their arguments to be *logical operands*. A *logical operand* is a [`Decimal`](#decimal.Decimal "decimal.Decimal") instance whose exponent and sign are both zero, and whose digits are all either `0` or `1`. ## Context objects Contexts are environments for arithmetic operations. They govern precision, set rules for rounding, determine which signals are treated as exceptions, and limit the range for exponents. Each thread has its own current context which is accessed or changed using the [`getcontext()`](#decimal.getcontext "decimal.getcontext") and [`setcontext()`](#decimal.setcontext "decimal.setcontext") functions: `decimal.``getcontext`()Return the current context for the active thread. `decimal.``setcontext`(*c*)Set the current context for the active thread to *c*. You can also use the [`with`](../reference/compound_stmts.xhtml#with) statement and the [`localcontext()`](#decimal.localcontext "decimal.localcontext")function to temporarily change the active context. `decimal.``localcontext`(*ctx=None*)Return a context manager that will set the current context for the active thread to a copy of *ctx* on entry to the with-statement and restore the previous context when exiting the with-statement. If no context is specified, a copy of the current context is used. For example, the following code sets the current decimal precision to 42 places, performs a calculation, and then automatically restores the previous context: ``` from decimal import localcontext with localcontext() as ctx: ctx.prec = 42 # Perform a high precision calculation s = calculate_something() s = +s # Round the final result back to the default precision ``` New contexts can also be created using the [`Context`](#decimal.Context "decimal.Context") constructor described below. In addition, the module provides three pre-made contexts: *class* `decimal.``BasicContext`This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to [`ROUND_HALF_UP`](#decimal.ROUND_HALF_UP "decimal.ROUND_HALF_UP"). All flags are cleared. All traps are enabled (treated as exceptions) except [`Inexact`](#decimal.Inexact "decimal.Inexact"), [`Rounded`](#decimal.Rounded "decimal.Rounded"), and [`Subnormal`](#decimal.Subnormal "decimal.Subnormal"). Because many of the traps are enabled, this context is useful for debugging. *class* `decimal.``ExtendedContext`This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to [`ROUND_HALF_EVEN`](#decimal.ROUND_HALF_EVEN "decimal.ROUND_HALF_EVEN"). All flags are cleared. No traps are enabled (so that exceptions are not raised during computations). Because the traps are disabled, this context is useful for applications that prefer to have result value of `NaN` or `Infinity` instead of raising exceptions. This allows an application to complete a run in the presence of conditions that would otherwise halt the program. *class* `decimal.``DefaultContext`This context is used by the [`Context`](#decimal.Context "decimal.Context") constructor as a prototype for new contexts. Changing a field (such a precision) has the effect of changing the default for new contexts created by the [`Context`](#decimal.Context "decimal.Context") constructor. This context is most useful in multi-threaded environments. Changing one of the fields before threads are started has the effect of setting system-wide defaults. Changing the fields after threads have started is not recommended as it would require thread synchronization to prevent race conditions. In single threaded environments, it is preferable to not use this context at all. Instead, simply create contexts explicitly as described below. The default values are `prec`=`28`, `rounding`=[`ROUND_HALF_EVEN`](#decimal.ROUND_HALF_EVEN "decimal.ROUND_HALF_EVEN"), and enabled traps for [`Overflow`](#decimal.Overflow "decimal.Overflow"), [`InvalidOperation`](#decimal.InvalidOperation "decimal.InvalidOperation"), and [`DivisionByZero`](#decimal.DivisionByZero "decimal.DivisionByZero"). In addition to the three supplied contexts, new contexts can be created with the [`Context`](#decimal.Context "decimal.Context") constructor. *class* `decimal.``Context`(*prec=None*, *rounding=None*, *Emin=None*, *Emax=None*, *capitals=None*, *clamp=None*, *flags=None*, *traps=None*)Creates a new context. If a field is not specified or is [`None`](constants.xhtml#None "None"), the default values are copied from the [`DefaultContext`](#decimal.DefaultContext "decimal.DefaultContext"). If the *flags*field is not specified or is [`None`](constants.xhtml#None "None"), all flags are cleared. *prec* is an integer in the range \[`1`, [`MAX_PREC`](#decimal.MAX_PREC "decimal.MAX_PREC")\] that sets the precision for arithmetic operations in the context. The *rounding* option is one of the constants listed in the section [Rounding Modes](#rounding-modes). The *traps* and *flags* fields list any signals to be set. Generally, new contexts should only set traps and leave the flags clear. The *Emin* and *Emax* fields are integers specifying the outer limits allowable for exponents. *Emin* must be in the range \[[`MIN_EMIN`](#decimal.MIN_EMIN "decimal.MIN_EMIN"), `0`\], *Emax* in the range \[`0`, [`MAX_EMAX`](#decimal.MAX_EMAX "decimal.MAX_EMAX")\]. The *capitals* field is either `0` or `1` (the default). If set to `1`, exponents are printed with a capital `E`; otherwise, a lowercase `e` is used: `Decimal('6.02e+23')`. The *clamp* field is either `0` (the default) or `1`. If set to `1`, the exponent `e` of a [`Decimal`](#decimal.Decimal "decimal.Decimal")instance representable in this context is strictly limited to the range `Emin - prec + 1 <= e <= Emax - prec + 1`. If *clamp* is `0` then a weaker condition holds: the adjusted exponent of the [`Decimal`](#decimal.Decimal "decimal.Decimal") instance is at most `Emax`. When *clamp* is `1`, a large normal number will, where possible, have its exponent reduced and a corresponding number of zeros added to its coefficient, in order to fit the exponent constraints; this preserves the value of the number but loses information about significant trailing zeros. For example: ``` >>> Context(prec=6, Emax=999, clamp=1).create_decimal('1.23e999') Decimal('1.23000E+999') ``` A *clamp* value of `1` allows compatibility with the fixed-width decimal interchange formats specified in IEEE 754. The [`Context`](#decimal.Context "decimal.Context") class defines several general purpose methods as well as a large number of methods for doing arithmetic directly in a given context. In addition, for each of the [`Decimal`](#decimal.Decimal "decimal.Decimal") methods described above (with the exception of the `adjusted()` and `as_tuple()` methods) there is a corresponding [`Context`](#decimal.Context "decimal.Context") method. For example, for a [`Context`](#decimal.Context "decimal.Context")instance `C` and [`Decimal`](#decimal.Decimal "decimal.Decimal") instance `x`, `C.exp(x)` is equivalent to `x.exp(context=C)`. Each [`Context`](#decimal.Context "decimal.Context") method accepts a Python integer (an instance of [`int`](functions.xhtml#int "int")) anywhere that a Decimal instance is accepted. `clear_flags`()Resets all of the flags to `0`. `clear_traps`()Resets all of the traps to `0`. 3\.3 新版功能. `copy`()Return a duplicate of the context. `copy_decimal`(*num*)Return a copy of the Decimal instance num. `create_decimal`(*num*)Creates a new Decimal instance from *num* but using *self* as context. Unlike the [`Decimal`](#decimal.Decimal "decimal.Decimal") constructor, the context precision, rounding method, flags, and traps are applied to the conversion. This is useful because constants are often given to a greater precision than is needed by the application. Another benefit is that rounding immediately eliminates unintended effects from digits beyond the current precision. In the following example, using unrounded inputs means that adding zero to a sum can change the result: ``` >>> getcontext().prec = 3 >>> Decimal('3.4445') + Decimal('1.0023') Decimal('4.45') >>> Decimal('3.4445') + Decimal(0) + Decimal('1.0023') Decimal('4.44') ``` This method implements the to-number operation of the IBM specification. If the argument is a string, no leading or trailing whitespace or underscores are permitted. `create_decimal_from_float`(*f*)Creates a new Decimal instance from a float *f* but rounding using *self*as the context. Unlike the [`Decimal.from_float()`](#decimal.Decimal.from_float "decimal.Decimal.from_float") class method, the context precision, rounding method, flags, and traps are applied to the conversion. ``` >>> context = Context(prec=5, rounding=ROUND_DOWN) >>> context.create_decimal_from_float(math.pi) Decimal('3.1415') >>> context = Context(prec=5, traps=[Inexact]) >>> context.create_decimal_from_float(math.pi) Traceback (most recent call last): ... decimal.Inexact: None ``` 3\.1 新版功能. `Etiny`()Returns a value equal to `Emin - prec + 1` which is the minimum exponent value for subnormal results. When underflow occurs, the exponent is set to [`Etiny`](#decimal.Context.Etiny "decimal.Context.Etiny"). `Etop`()Returns a value equal to `Emax - prec + 1`. The usual approach to working with decimals is to create [`Decimal`](#decimal.Decimal "decimal.Decimal")instances and then apply arithmetic operations which take place within the current context for the active thread. An alternative approach is to use context methods for calculating within a specific context. The methods are similar to those for the [`Decimal`](#decimal.Decimal "decimal.Decimal") class and are only briefly recounted here. `abs`(*x*)Returns the absolute value of *x*. `add`(*x*, *y*)Return the sum of *x* and *y*. `canonical`(*x*)Returns the same Decimal object *x*. `compare`(*x*, *y*)Compares *x* and *y* numerically. `compare_signal`(*x*, *y*)Compares the values of the two operands numerically. `compare_total`(*x*, *y*)Compares two operands using their abstract representation. `compare_total_mag`(*x*, *y*)Compares two operands using their abstract representation, ignoring sign. `copy_abs`(*x*)Returns a copy of *x* with the sign set to 0. `copy_negate`(*x*)Returns a copy of *x* with the sign inverted. `copy_sign`(*x*, *y*)Copies the sign from *y* to *x*. `divide`(*x*, *y*)Return *x* divided by *y*. `divide_int`(*x*, *y*)Return *x* divided by *y*, truncated to an integer. `divmod`(*x*, *y*)Divides two numbers and returns the integer part of the result. `exp`(*x*)Returns e \*\* x. `fma`(*x*, *y*, *z*)Returns *x* multiplied by *y*, plus *z*. `is_canonical`(*x*)Returns `True` if *x* is canonical; otherwise returns `False`. `is_finite`(*x*)Returns `True` if *x* is finite; otherwise returns `False`. `is_infinite`(*x*)Returns `True` if *x* is infinite; otherwise returns `False`. `is_nan`(*x*)Returns `True` if *x* is a qNaN or sNaN; otherwise returns `False`. `is_normal`(*x*)Returns `True` if *x* is a normal number; otherwise returns `False`. `is_qnan`(*x*)Returns `True` if *x* is a quiet NaN; otherwise returns `False`. `is_signed`(*x*)Returns `True` if *x* is negative; otherwise returns `False`. `is_snan`(*x*)Returns `True` if *x* is a signaling NaN; otherwise returns `False`. `is_subnormal`(*x*)Returns `True` if *x* is subnormal; otherwise returns `False`. `is_zero`(*x*)Returns `True` if *x* is a zero; otherwise returns `False`. `ln`(*x*)Returns the natural (base e) logarithm of *x*. `log10`(*x*)Returns the base 10 logarithm of *x*. `logb`(*x*)Returns the exponent of the magnitude of the operand's MSD. `logical_and`(*x*, *y*)Applies the logical operation *and* between each operand's digits. `logical_invert`(*x*)Invert all the digits in *x*. `logical_or`(*x*, *y*)Applies the logical operation *or* between each operand's digits. `logical_xor`(*x*, *y*)Applies the logical operation *xor* between each operand's digits. `max`(*x*, *y*)Compares two values numerically and returns the maximum. `max_mag`(*x*, *y*)Compares the values numerically with their sign ignored. `min`(*x*, *y*)Compares two values numerically and returns the minimum. `min_mag`(*x*, *y*)Compares the values numerically with their sign ignored. `minus`(*x*)Minus corresponds to the unary prefix minus operator in Python. `multiply`(*x*, *y*)Return the product of *x* and *y*. `next_minus`(*x*)Returns the largest representable number smaller than *x*. `next_plus`(*x*)Returns the smallest representable number larger than *x*. `next_toward`(*x*, *y*)Returns the number closest to *x*, in direction towards *y*. `normalize`(*x*)Reduces *x* to its simplest form. `number_class`(*x*)Returns an indication of the class of *x*. `plus`(*x*)Plus corresponds to the unary prefix plus operator in Python. This operation applies the context precision and rounding, so it is *not* an identity operation. `power`(*x*, *y*, *modulo=None*)Return `x` to the power of `y`, reduced modulo `modulo` if given. With two arguments, compute `x**y`. If `x` is negative then `y`must be integral. The result will be inexact unless `y` is integral and the result is finite and can be expressed exactly in 'precision' digits. The rounding mode of the context is used. Results are always correctly-rounded in the Python version. 在 3.3 版更改: The C module computes [`power()`](#decimal.Context.power "decimal.Context.power") in terms of the correctly-rounded [`exp()`](#decimal.Context.exp "decimal.Context.exp") and [`ln()`](#decimal.Context.ln "decimal.Context.ln") functions. The result is well-defined but only "almost always correctly-rounded". With three arguments, compute `(x**y) % modulo`. For the three argument form, the following restrictions on the arguments hold: > - all three arguments must be integral > - `y` must be nonnegative > - at least one of `x` or `y` must be nonzero > - `modulo` must be nonzero and have at most 'precision' digits The value resulting from `Context.power(x, y, modulo)` is equal to the value that would be obtained by computing ``` (x**y) % modulo ``` with unbounded precision, but is computed more efficiently. The exponent of the result is zero, regardless of the exponents of `x`, `y` and `modulo`. The result is always exact. `quantize`(*x*, *y*)Returns a value equal to *x* (rounded), having the exponent of *y*. `radix`()Just returns 10, as this is Decimal, :) `remainder`(*x*, *y*)Returns the remainder from integer division. The sign of the result, if non-zero, is the same as that of the original dividend. `remainder_near`(*x*, *y*)Returns `x - y * n`, where *n* is the integer nearest the exact value of `x / y` (if the result is 0 then its sign will be the sign of *x*). `rotate`(*x*, *y*)Returns a rotated copy of *x*, *y* times. `same_quantum`(*x*, *y*)Returns `True` if the two operands have the same exponent. `scaleb`(*x*, *y*)Returns the first operand after adding the second value its exp. `shift`(*x*, *y*)Returns a shifted copy of *x*, *y* times. `sqrt`(*x*)Square root of a non-negative number to context precision. `subtract`(*x*, *y*)Return the difference between *x* and *y*. `to_eng_string`(*x*)Convert to a string, using engineering notation if an exponent is needed. Engineering notation has an exponent which is a multiple of 3. This can leave up to 3 digits to the left of the decimal place and may require the addition of either one or two trailing zeros. `to_integral_exact`(*x*)Rounds to an integer. `to_sci_string`(*x*)Converts a number to a string using scientific notation. ## 常數 The constants in this section are only relevant for the C module. They are also included in the pure Python version for compatibility. 32-bit 64-bit `decimal.``MAX_PREC``425000000` `999999999999999999` `decimal.``MAX_EMAX``425000000` `999999999999999999` `decimal.``MIN_EMIN``-425000000` `-999999999999999999` `decimal.``MIN_ETINY``-849999999` `-1999999999999999997` `decimal.``HAVE_THREADS`The default value is `True`. If Python is compiled without threads, the C version automatically disables the expensive thread local context machinery. In this case, the value is `False`. ## Rounding modes `decimal.``ROUND_CEILING`Round towards `Infinity`. `decimal.``ROUND_DOWN`Round towards zero. `decimal.``ROUND_FLOOR`Round towards `-Infinity`. `decimal.``ROUND_HALF_DOWN`Round to nearest with ties going towards zero. `decimal.``ROUND_HALF_EVEN`Round to nearest with ties going to nearest even integer. `decimal.``ROUND_HALF_UP`Round to nearest with ties going away from zero. `decimal.``ROUND_UP`Round away from zero. `decimal.``ROUND_05UP`Round away from zero if last digit after rounding towards zero would have been 0 or 5; otherwise round towards zero. ## Signals Signals represent conditions that arise during computation. Each corresponds to one context flag and one context trap enabler. The context flag is set whenever the condition is encountered. After the computation, flags may be checked for informational purposes (for instance, to determine whether a computation was exact). After checking the flags, be sure to clear all flags before starting the next computation. If the context's trap enabler is set for the signal, then the condition causes a Python exception to be raised. For example, if the [`DivisionByZero`](#decimal.DivisionByZero "decimal.DivisionByZero") trap is set, then a [`DivisionByZero`](#decimal.DivisionByZero "decimal.DivisionByZero") exception is raised upon encountering the condition. *class* `decimal.``Clamped`Altered an exponent to fit representation constraints. Typically, clamping occurs when an exponent falls outside the context's `Emin` and `Emax` limits. If possible, the exponent is reduced to fit by adding zeros to the coefficient. *class* `decimal.``DecimalException`Base class for other signals and a subclass of [`ArithmeticError`](exceptions.xhtml#ArithmeticError "ArithmeticError"). *class* `decimal.``DivisionByZero`Signals the division of a non-infinite number by zero. Can occur with division, modulo division, or when raising a number to a negative power. If this signal is not trapped, returns `Infinity` or `-Infinity` with the sign determined by the inputs to the calculation. *class* `decimal.``Inexact`Indicates that rounding occurred and the result is not exact. Signals when non-zero digits were discarded during rounding. The rounded result is returned. The signal flag or trap is used to detect when results are inexact. *class* `decimal.``InvalidOperation`An invalid operation was performed. Indicates that an operation was requested that does not make sense. If not trapped, returns `NaN`. Possible causes include: ``` Infinity - Infinity 0 * Infinity Infinity / Infinity x % 0 Infinity % x sqrt(-x) and x > 0 0 ** 0 x ** (non-integer) x ** Infinity ``` *class* `decimal.``Overflow`Numerical overflow. Indicates the exponent is larger than `Emax` after rounding has occurred. If not trapped, the result depends on the rounding mode, either pulling inward to the largest representable finite number or rounding outward to `Infinity`. In either case, [`Inexact`](#decimal.Inexact "decimal.Inexact") and [`Rounded`](#decimal.Rounded "decimal.Rounded")are also signaled. *class* `decimal.``Rounded`Rounding occurred though possibly no information was lost. Signaled whenever rounding discards digits; even if those digits are zero (such as rounding `5.00` to `5.0`). If not trapped, returns the result unchanged. This signal is used to detect loss of significant digits. *class* `decimal.``Subnormal`Exponent was lower than `Emin` prior to rounding. Occurs when an operation result is subnormal (the exponent is too small). If not trapped, returns the result unchanged. *class* `decimal.``Underflow`Numerical underflow with result rounded to zero. Occurs when a subnormal result is pushed to zero by rounding. [`Inexact`](#decimal.Inexact "decimal.Inexact")and [`Subnormal`](#decimal.Subnormal "decimal.Subnormal") are also signaled. *class* `decimal.``FloatOperation`Enable stricter semantics for mixing floats and Decimals. If the signal is not trapped (default), mixing floats and Decimals is permitted in the [`Decimal`](#decimal.Decimal "decimal.Decimal") constructor, [`create_decimal()`](#decimal.Context.create_decimal "decimal.Context.create_decimal") and all comparison operators. Both conversion and comparisons are exact. Any occurrence of a mixed operation is silently recorded by setting [`FloatOperation`](#decimal.FloatOperation "decimal.FloatOperation") in the context flags. Explicit conversions with [`from_float()`](#decimal.Decimal.from_float "decimal.Decimal.from_float")or [`create_decimal_from_float()`](#decimal.Context.create_decimal_from_float "decimal.Context.create_decimal_from_float") do not set the flag. Otherwise (the signal is trapped), only equality comparisons and explicit conversions are silent. All other mixed operations raise [`FloatOperation`](#decimal.FloatOperation "decimal.FloatOperation"). The following table summarizes the hierarchy of signals: ``` exceptions.ArithmeticError(exceptions.Exception) DecimalException Clamped DivisionByZero(DecimalException, exceptions.ZeroDivisionError) Inexact Overflow(Inexact, Rounded) Underflow(Inexact, Rounded, Subnormal) InvalidOperation Rounded Subnormal FloatOperation(DecimalException, exceptions.TypeError) ``` ## Floating Point Notes ### Mitigating round-off error with increased precision The use of decimal floating point eliminates decimal representation error (making it possible to represent `0.1` exactly); however, some operations can still incur round-off error when non-zero digits exceed the fixed precision. The effects of round-off error can be amplified by the addition or subtraction of nearly offsetting quantities resulting in loss of significance. Knuth provides two instructive examples where rounded floating point arithmetic with insufficient precision causes the breakdown of the associative and distributive properties of addition: ``` # Examples from Seminumerical Algorithms, Section 4.2.2. >>> from decimal import Decimal, getcontext >>> getcontext().prec = 8 >>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111') >>> (u + v) + w Decimal('9.5111111') >>> u + (v + w) Decimal('10') >>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003') >>> (u*v) + (u*w) Decimal('0.01') >>> u * (v+w) Decimal('0.0060000') ``` The [`decimal`](#module-decimal "decimal: Implementation of the General Decimal Arithmetic Specification.") module makes it possible to restore the identities by expanding the precision sufficiently to avoid loss of significance: ``` >>> getcontext().prec = 20 >>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111') >>> (u + v) + w Decimal('9.51111111') >>> u + (v + w) Decimal('9.51111111') >>> >>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003') >>> (u*v) + (u*w) Decimal('0.0060000') >>> u * (v+w) Decimal('0.0060000') ``` ### Special values The number system for the [`decimal`](#module-decimal "decimal: Implementation of the General Decimal Arithmetic Specification.") module provides special values including `NaN`, `sNaN`, `-Infinity`, `Infinity`, and two zeros, `+0` and `-0`. Infinities can be constructed directly with: `Decimal('Infinity')`. Also, they can arise from dividing by zero when the [`DivisionByZero`](#decimal.DivisionByZero "decimal.DivisionByZero") signal is not trapped. Likewise, when the [`Overflow`](#decimal.Overflow "decimal.Overflow") signal is not trapped, infinity can result from rounding beyond the limits of the largest representable number. The infinities are signed (affine) and can be used in arithmetic operations where they get treated as very large, indeterminate numbers. For instance, adding a constant to infinity gives another infinite result. Some operations are indeterminate and return `NaN`, or if the [`InvalidOperation`](#decimal.InvalidOperation "decimal.InvalidOperation") signal is trapped, raise an exception. For example, `0/0` returns `NaN` which means "not a number". This variety of `NaN` is quiet and, once created, will flow through other computations always resulting in another `NaN`. This behavior can be useful for a series of computations that occasionally have missing inputs --- it allows the calculation to proceed while flagging specific results as invalid. A variant is `sNaN` which signals rather than remaining quiet after every operation. This is a useful return value when an invalid result needs to interrupt a calculation for special handling. The behavior of Python's comparison operators can be a little surprising where a `NaN` is involved. A test for equality where one of the operands is a quiet or signaling `NaN` always returns [`False`](constants.xhtml#False "False") (even when doing `Decimal('NaN')==Decimal('NaN')`), while a test for inequality always returns [`True`](constants.xhtml#True "True"). An attempt to compare two Decimals using any of the `<`, `<=`, `>` or `>=` operators will raise the [`InvalidOperation`](#decimal.InvalidOperation "decimal.InvalidOperation") signal if either operand is a `NaN`, and return [`False`](constants.xhtml#False "False") if this signal is not trapped. Note that the General Decimal Arithmetic specification does not specify the behavior of direct comparisons; these rules for comparisons involving a `NaN` were taken from the IEEE 854 standard (see Table 3 in section 5.7). To ensure strict standards-compliance, use the `compare()`and `compare-signal()` methods instead. The signed zeros can result from calculations that underflow. They keep the sign that would have resulted if the calculation had been carried out to greater precision. Since their magnitude is zero, both positive and negative zeros are treated as equal and their sign is informational. In addition to the two signed zeros which are distinct yet equal, there are various representations of zero with differing precisions yet equivalent in value. This takes a bit of getting used to. For an eye accustomed to normalized floating point representations, it is not immediately obvious that the following calculation returns a value equal to zero: ``` >>> 1 / Decimal('Infinity') Decimal('0E-1000026') ``` ## Working with threads The [`getcontext()`](#decimal.getcontext "decimal.getcontext") function accesses a different [`Context`](#decimal.Context "decimal.Context") object for each thread. Having separate thread contexts means that threads may make changes (such as `getcontext().prec=10`) without interfering with other threads. Likewise, the [`setcontext()`](#decimal.setcontext "decimal.setcontext") function automatically assigns its target to the current thread. If [`setcontext()`](#decimal.setcontext "decimal.setcontext") has not been called before [`getcontext()`](#decimal.getcontext "decimal.getcontext"), then [`getcontext()`](#decimal.getcontext "decimal.getcontext") will automatically create a new context for use in the current thread. The new context is copied from a prototype context called *DefaultContext*. To control the defaults so that each thread will use the same values throughout the application, directly modify the *DefaultContext* object. This should be done *before* any threads are started so that there won't be a race condition between threads calling [`getcontext()`](#decimal.getcontext "decimal.getcontext"). For example: ``` # Set applicationwide defaults for all threads about to be launched DefaultContext.prec = 12 DefaultContext.rounding = ROUND_DOWN DefaultContext.traps = ExtendedContext.traps.copy() DefaultContext.traps[InvalidOperation] = 1 setcontext(DefaultContext) # Afterwards, the threads can be started t1.start() t2.start() t3.start() . . . ``` ## Recipes Here are a few recipes that serve as utility functions and that demonstrate ways to work with the [`Decimal`](#decimal.Decimal "decimal.Decimal") class: ``` def moneyfmt(value, places=2, curr='', sep=',', dp='.', pos='', neg='-', trailneg=''): """Convert Decimal to a money formatted string. places: required number of places after the decimal point curr: optional currency symbol before the sign (may be blank) sep: optional grouping separator (comma, period, space, or blank) dp: decimal point indicator (comma or period) only specify as blank when places is zero pos: optional sign for positive numbers: '+', space or blank neg: optional sign for negative numbers: '-', '(', space or blank trailneg:optional trailing minus indicator: '-', ')', space or blank >>> d = Decimal('-1234567.8901') >>> moneyfmt(d, curr='$') '-$1,234,567.89' >>> moneyfmt(d, places=0, sep='.', dp='', neg='', trailneg='-') '1.234.568-' >>> moneyfmt(d, curr='$', neg='(', trailneg=')') '($1,234,567.89)' >>> moneyfmt(Decimal(123456789), sep=' ') '123 456 789.00' >>> moneyfmt(Decimal('-0.02'), neg='<', trailneg='>') '<0.02>' """ q = Decimal(10) ** -places # 2 places --> '0.01' sign, digits, exp = value.quantize(q).as_tuple() result = [] digits = list(map(str, digits)) build, next = result.append, digits.pop if sign: build(trailneg) for i in range(places): build(next() if digits else '0') if places: build(dp) if not digits: build('0') i = 0 while digits: build(next()) i += 1 if i == 3 and digits: i = 0 build(sep) build(curr) build(neg if sign else pos) return ''.join(reversed(result)) def pi(): """Compute Pi to the current precision. >>> print(pi()) 3.141592653589793238462643383 """ getcontext().prec += 2 # extra digits for intermediate steps three = Decimal(3) # substitute "three=3.0" for regular floats lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24 while s != lasts: lasts = s n, na = n+na, na+8 d, da = d+da, da+32 t = (t * n) / d s += t getcontext().prec -= 2 return +s # unary plus applies the new precision def exp(x): """Return e raised to the power of x. Result type matches input type. >>> print(exp(Decimal(1))) 2.718281828459045235360287471 >>> print(exp(Decimal(2))) 7.389056098930650227230427461 >>> print(exp(2.0)) 7.38905609893 >>> print(exp(2+0j)) (7.38905609893+0j) """ getcontext().prec += 2 i, lasts, s, fact, num = 0, 0, 1, 1, 1 while s != lasts: lasts = s i += 1 fact *= i num *= x s += num / fact getcontext().prec -= 2 return +s def cos(x): """Return the cosine of x as measured in radians. The Taylor series approximation works best for a small value of x. For larger values, first compute x = x % (2 * pi). >>> print(cos(Decimal('0.5'))) 0.8775825618903727161162815826 >>> print(cos(0.5)) 0.87758256189 >>> print(cos(0.5+0j)) (0.87758256189+0j) """ getcontext().prec += 2 i, lasts, s, fact, num, sign = 0, 0, 1, 1, 1, 1 while s != lasts: lasts = s i += 2 fact *= i * (i-1) num *= x * x sign *= -1 s += num / fact * sign getcontext().prec -= 2 return +s def sin(x): """Return the sine of x as measured in radians. The Taylor series approximation works best for a small value of x. For larger values, first compute x = x % (2 * pi). >>> print(sin(Decimal('0.5'))) 0.4794255386042030002732879352 >>> print(sin(0.5)) 0.479425538604 >>> print(sin(0.5+0j)) (0.479425538604+0j) """ getcontext().prec += 2 i, lasts, s, fact, num, sign = 1, 0, x, 1, x, 1 while s != lasts: lasts = s i += 2 fact *= i * (i-1) num *= x * x sign *= -1 s += num / fact * sign getcontext().prec -= 2 return +s ``` ## Decimal FAQ Q. It is cumbersome to type `decimal.Decimal('1234.5')`. Is there a way to minimize typing when using the interactive interpreter? A. Some users abbreviate the constructor to just a single letter: ``` >>> D = decimal.Decimal >>> D('1.23') + D('3.45') Decimal('4.68') ``` Q. In a fixed-point application with two decimal places, some inputs have many places and need to be rounded. Others are not supposed to have excess digits and need to be validated. What methods should be used? A. The `quantize()` method rounds to a fixed number of decimal places. If the [`Inexact`](#decimal.Inexact "decimal.Inexact") trap is set, it is also useful for validation: ``` >>> TWOPLACES = Decimal(10) ** -2 # same as Decimal('0.01') ``` ``` >>> # Round to two places >>> Decimal('3.214').quantize(TWOPLACES) Decimal('3.21') ``` ``` >>> # Validate that a number does not exceed two places >>> Decimal('3.21').quantize(TWOPLACES, context=Context(traps=[Inexact])) Decimal('3.21') ``` ``` >>> Decimal('3.214').quantize(TWOPLACES, context=Context(traps=[Inexact])) Traceback (most recent call last): ... Inexact: None ``` Q. Once I have valid two place inputs, how do I maintain that invariant throughout an application? A. Some operations like addition, subtraction, and multiplication by an integer will automatically preserve fixed point. Others operations, like division and non-integer multiplication, will change the number of decimal places and need to be followed-up with a `quantize()` step: ``` >>> a = Decimal('102.72') # Initial fixed-point values >>> b = Decimal('3.17') >>> a + b # Addition preserves fixed-point Decimal('105.89') >>> a - b Decimal('99.55') >>> a * 42 # So does integer multiplication Decimal('4314.24') >>> (a * b).quantize(TWOPLACES) # Must quantize non-integer multiplication Decimal('325.62') >>> (b / a).quantize(TWOPLACES) # And quantize division Decimal('0.03') ``` In developing fixed-point applications, it is convenient to define functions to handle the `quantize()` step: ``` >>> def mul(x, y, fp=TWOPLACES): ... return (x * y).quantize(fp) >>> def div(x, y, fp=TWOPLACES): ... return (x / y).quantize(fp) ``` ``` >>> mul(a, b) # Automatically preserve fixed-point Decimal('325.62') >>> div(b, a) Decimal('0.03') ``` Q. There are many ways to express the same value. The numbers `200`, `200.000`, `2E2`, and `02E+4` all have the same value at various precisions. Is there a way to transform them to a single recognizable canonical value? A. The `normalize()` method maps all equivalent values to a single representative: ``` >>> values = map(Decimal, '200 200.000 2E2 .02E+4'.split()) >>> [v.normalize() for v in values] [Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2')] ``` Q. Some decimal values always print with exponential notation. Is there a way to get a non-exponential representation? A. For some values, exponential notation is the only way to express the number of significant places in the coefficient. For example, expressing `5.0E+3` as `5000` keeps the value constant but cannot show the original's two-place significance. If an application does not care about tracking significance, it is easy to remove the exponent and trailing zeroes, losing significance, but keeping the value unchanged: ``` >>> def remove_exponent(d): ... return d.quantize(Decimal(1)) if d == d.to_integral() else d.normalize() ``` ``` >>> remove_exponent(Decimal('5E+3')) Decimal('5000') ``` Q. Is there a way to convert a regular float to a [`Decimal`](#decimal.Decimal "decimal.Decimal")? A. Yes, any binary floating point number can be exactly expressed as a Decimal though an exact conversion may take more precision than intuition would suggest: ``` >>> Decimal(math.pi) Decimal('3.141592653589793115997963468544185161590576171875') ``` Q. Within a complex calculation, how can I make sure that I haven't gotten a spurious result because of insufficient precision or rounding anomalies. A. The decimal module makes it easy to test results. A best practice is to re-run calculations using greater precision and with various rounding modes. Widely differing results indicate insufficient precision, rounding mode issues, ill-conditioned inputs, or a numerically unstable algorithm. Q. I noticed that context precision is applied to the results of operations but not to the inputs. Is there anything to watch out for when mixing values of different precisions? A. Yes. The principle is that all values are considered to be exact and so is the arithmetic on those values. Only the results are rounded. The advantage for inputs is that "what you type is what you get". A disadvantage is that the results can look odd if you forget that the inputs haven't been rounded: ``` >>> getcontext().prec = 3 >>> Decimal('3.104') + Decimal('2.104') Decimal('5.21') >>> Decimal('3.104') + Decimal('0.000') + Decimal('2.104') Decimal('5.20') ``` The solution is either to increase precision or to force rounding of inputs using the unary plus operation: ``` >>> getcontext().prec = 3 >>> +Decimal('1.23456789') # unary plus triggers rounding Decimal('1.23') ``` Alternatively, inputs can be rounded upon creation using the [`Context.create_decimal()`](#decimal.Context.create_decimal "decimal.Context.create_decimal") method: ``` >>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678') Decimal('1.2345') ``` Q. Is the CPython implementation fast for large numbers? A. Yes. In the CPython and PyPy3 implementations, the C/CFFI versions of the decimal module integrate the high speed [libmpdec](https://www.bytereef.org/mpdecimal/doc/libmpdec/index.html) \[https://www.bytereef.org/mpdecimal/doc/libmpdec/index.html\] library for arbitrary precision correctly-rounded decimal floating point arithmetic. `libmpdec` uses [Karatsuba multiplication](https://en.wikipedia.org/wiki/Karatsuba_algorithm) \[https://en.wikipedia.org/wiki/Karatsuba\_algorithm\]for medium-sized numbers and the [Number Theoretic Transform](https://en.wikipedia.org/wiki/Discrete_Fourier_transform_(general)#Number-theoretic_transform) \[https://en.wikipedia.org/wiki/Discrete\_Fourier\_transform\_(general)#Number-theoretic\_transform\]for very large numbers. However, to realize this performance gain, the context needs to be set for unrounded calculations. ``` >>> c = getcontext() >>> c.prec = MAX_PREC >>> c.Emax = MAX_EMAX >>> c.Emin = MIN_EMIN ``` 3\.3 新版功能. ### 導航 - [索引](../genindex.xhtml "總目錄") - [模塊](../py-modindex.xhtml "Python 模塊索引") | - [下一頁](fractions.xhtml "fractions --- 分數") | - [上一頁](cmath.xhtml "cmath --- Mathematical functions for complex numbers") | - ![](https://box.kancloud.cn/a721fc7ec672275e257bbbfde49a4d4e_16x16.png) - [Python](https://www.python.org/) ? - zh\_CN 3.7.3 [文檔](../index.xhtml) ? - [Python 標準庫](index.xhtml) ? - [數字和數學模塊](numeric.xhtml) ? - $('.inline-search').show(0); | ? [版權所有](../copyright.xhtml) 2001-2019, Python Software Foundation. Python 軟件基金會是一個非盈利組織。 [請捐助。](https://www.python.org/psf/donations/) 最后更新于 5月 21, 2019. [發現了問題](../bugs.xhtml)? 使用[Sphinx](http://sphinx.pocoo.org/)1.8.4 創建。
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