除以零

维基百科，自由的百科全书

此條目仍有文字未被翻譯成中文，條目是根據其他語言維基百科的內容進行翻譯的。（2008年4月23日）
歡迎您协助翻译与校对以改善这篇條目。
长期闲置的非中文内容可能会被移除。

數學中，將某數除以零可表達為 $\textstyle\frac{a}{0}$ ，即a除以零。此式是否成立端視其在如何的數學設定下計算。一般實數算術中，此式為無意義。在程序設計中，當遇上正整數除以零程序會中止，正如浮點數會出現NaN值的情況。

[编辑] 基本算術

基本算術中，除法指將一個集合中的物件分成若干等份。例如，10個蘋果平分給5人，每人可得 $\textstyle\frac{10}{5}$ = 2個蘋果。同理，10個蘋果只分給1人，則他/她可得 $\textstyle\frac{10}{1}$ = 10個蘋果。

若除以0又如何？若有10個蘋果，無人來分，每「人」可得多少蘋果？問題本身是沒有意義的，根本無人來，談論每「人」可得多少根本多餘。所以， $\textstyle\frac{10}{0}$ ，在基本算術中，是無意義或未下定義的。

另一種解釋是將除法理解為不斷的減法。例如「13除以5」，換一種說法，13減去兩個5，餘下3，即被除數一直減去除數直至餘數數值低於被除數，算式為 $\textstyle\frac{13}{5}$ = 2 餘數 3。若某數除以零，就算不斷減去零，餘數也不可能小於被除數，使得算式與無窮拉上關係，超出基本算術的範疇。

[编辑] 早期嘗試

婆羅摩笈多（598–668年）的著作Brahmasphutasiddhanta被視為最早討論零的數學和定義涉及零的算式的文本。但當中對除以零的論述並不正確，根據婆羅摩笈多，

"一個正或負整數除以零，成為以零為分母的分數。零除以正或負整數是零或以零為分子、該正或負整數為分母的分數。零除以零是零。"

830年，摩訶吠羅在其著作Ganita Sara Samgraha試圖糾正婆羅摩笈多的錯誤，但不成功：

"一數字除以零會維持不變。"

婆什迦羅第二嘗試解決此問題，設 $\textstyle\frac{n}{0}=\infty$ ，雖然此定義有一定道理，但會導致悖論^[1]（參見下面）。

[编辑] 代數處理

若某數學系統遵從域的公理，則在該數學系統內除以零必須為沒有意義。這是因為除法被定義為是乘法的逆向操作，即 $\textstyle\frac{a}{b}$ 值是方程 $b x = a$ 中x的解（若有的話）。若設b = 0，方程式bx = a可寫成 0x = a或直接 0 = a。因此，方程式bx = a沒有解若a不等於0，但x是任何數值也可解此方程若a等於0。在各自情況下均沒有獨一無二的數值，所以 $\textstyle\frac{a}{b}$ 未能下定義。

[编辑] 除以零的謬誤

在代數運算中不當使用除以零可得出無效證明：2 = 1

由：

$0\times 1 = 0$

$0\times 2 = 0$

得出：

$0\times 1 = 0\times 2$

除以零得出

$\textstyle \frac{0}{0}\times 1 = \frac{0}{0}\times 2$

簡化，得出：

$1 = 2\,$

以上謬論一個假設，就是某數除以0是容許的並且 $0 / 0 = 1$ 。

[编辑] 虛假的除法

在矩陣代數或線性代數中，可定義一種虛假的除法，設 $\textstyle\frac{a}{b}=a b^+$ ，當中b⁺代表b的虛構倒數。這樣，若b⁻¹存在，則b⁺ = b⁻¹。若b等於0，則0⁺ = 0；參見Generalized inverse。

[编辑] 抽象代數

The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication, so as above, division poses problems only when attempting to divide by zero. This is likewise true in a skew field (which for this reason is called a division ring). However, in other rings, division by nonzero elements may also pose problems. For example, the ring Z/6Z of integers mod 6. The meaning of the expression $\textstyle\frac{2}{2}$ should be the solution x of the equation $2 x = 2$ . But in the ring Z/6Z, 2 is not invertible under multiplication. This equation has two distinct solutions, x = 1 and x = 4, so the expression $\textstyle\frac{2}{2}$ is undefined.

In field theory, the expression $\textstyle\frac{a}{b}$ is only shorthand for the formal expression $a b - 1$ , where $b - 1$ is the multiplicative inverse of $b$ . Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when $b$ is zero. In modern texts the axiom $\textstyle 0\neq 1$ is included in order to avoid having to consider the one-element field where the multiplicative identity coincides with the additive identity. In such 'fields' however, $0 = 1$ , and $\textstyle\frac{0}{0}=\frac{0}{1}=0$ , and division by zero is actually noncontradictory.

[编辑] 數學分析

函數 y = $\textstyle\frac{1}{x}$ 。當 x 趨向 0， y 趨向無限（反之亦然）

[编辑] 扩展的实数轴

表面看來，可以藉着考慮隨着b趨向0的 $\textstyle\frac{a}{b}$ 極限而定義 $\textstyle\frac{a}{0}$ 。對於任何正數a，

$\lim_{b \to 0^{+}} {a \over b} = {+}\infty$

而對於任何負數a，

$\lim_{b \to 0^{+}} {a \over b} = {-}\infty.$

所以，對於正數a， $\textstyle\frac{a}{0}$ 可被定義為+∞，而對於負數a則可定義為−∞。不過，某數也可以由負數一方（左面）趨向零，這様，對於正數a， $\textstyle\frac{a}{0}$ 定義為−∞，負數a定義為+∞。由此可得（假設實數的基本性質可應用在極限上）：

$+\infty = \frac{1}{0} = \frac{1}{-0} = -\frac{1}{0} = -\infty$

最終變成 +∞ = −∞，與在扩展的实数轴上對極限賦予的標準定義不相符。唯一的辦法是用沒有正負號的無限，參見下面。

另外，利用極限的比無為 $\textstyle\frac{0}{0}$ 提供解釋：

$\lim_{(a,b) \to (0,0)} {a \over b}$

並不存在，而

$\lim_{x \to 0} {f(x) \over g(x)}$

若隨着x趨向0，f(x)與g(x)均趨向0，該極限可等於任何實數或無限，或者根本不存在，視乎f及g是何函數（參閱洛必達法則）。由此， $\textstyle\frac{0}{0}$ 難以被定義為一極限。

[编辑] 形式推算

運用形式推算（formal calculation），正號、負號或沒有正負號因情況而定，除以零定義為：

$\lim\limits_{x \to 0} {\frac{1}{x} =\frac{\lim\limits_{x \to 0} {1}}{\lim\limits_{x \to 0} {x}}} = \frac{1}{0} = \infty.$

[编辑] Real projective line

The set $\mathbb{R}\cup\{\infty\}$ is the real projective line, which is a one-point compactification of the real line. Here $\infty$ means an unsigned infinity, an infinite quantity which is neither positive nor negative. This quantity satisfies $-\infty = \infty$ which, as we have seen, is necessary in this context. In this structure, we can define $\textstyle\frac{a}{0} = \infty$ for nonzero a, and $\textstyle\frac{a}{\infty} = 0$ . It is the natural way to view the range of the tangent and cotangent functions of trigonometry: tan(x) approaches the single point at infinity as x approaches either $\textstyle+\frac{\pi}{2}$ or $\textstyle-\frac{\pi}{2}$ from either direction.

This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, $\infty + \infty$ has no meaning in the projective line.

[编辑] 黎曼球

集合 $\mathbb{C}\cup\{\infty\}$ 為黎曼球（Riemann sphere），在複分析中相當重要。Here, too, $\infty$ is an unsigned infinity, or, as it is often called in this context, the point at infinity. This set is analogous to the real projective line, except that it is based on the field of complex numbers. In the Riemann sphere, $1/0=\infty$ , but $0 / 0$ is undefined, as well as $0\infty$ .

[编辑] Extended non-negative real number line

The negative real numbers can be discarded, and infinity introduced, leading to the set $[0, \infty]$ , where division by zero can be naturally defined as $\textstyle\frac{a}{0} = \infty$ for positive a. While this makes division defined in more cases than usual, subtraction is instead left undefined in many cases, because there are no negative numbers.

[编辑] Distribution theory

In distribution theory one can extend the function $\textstyle\frac{1}{x}$ to a distribution on the whole space of real numbers (in effect by using Cauchy principal values). It does not, however, make sense to ask for a 'value' of this distribution at $x = 0$ ; a sophisticated answer refers to the singular support of the distribution.

[编辑] Other number systems

Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define division by zero in other mathematical structures.

[编辑] Non-standard analysis

In the hyperreal numbers and the surreal numbers, division by zero is still impossible, but division by non-zero infinitesimals is possible.

[编辑] Abstract algebra

Any number system which forms a commutative ring, as do the integers, the real numbers, and the complex numbers, for instance, can be extended to a wheel in which division by zero is always possible, but division has then a slightly different meaning.

[编辑] Division by zero in computer arithmetic

The IEEE floating-point standard, supported by almost all modern processors, specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. In IEEE 754 arithmetic, a ÷ 0 is positive infinity when a is positive, negative infinity when a is negative, and NaN (not a number) when a = 0. The infinity signs change when dividing by −0 instead. This is possible because in IEEE 754 there are two zero values, plus zero and minus zero, and thus no ambiguity.

Integer division by zero is usually handled differently from floating point since there is no integer representation for the result. Some processors generate an exception when an attempt is made to divide an integer by zero, although others will simply continue and generate an incorrect result for the division. (That result is often zero.)

Because of the improper algebraic results of assigning any value to division by zero, many computer programming languages (including those used by calculators) explicitly forbid the execution of the operation and may prematurely halt a program that attempts it, sometimes reporting a "Divide by zero" error. Some programs (especially those that use fixed-point arithmetic where no dedicated floating-point hardware is available) will use behavior similar to the IEEE standard, using large positive and negative numbers to approximate infinities. In some programming languages, an attempt to divide by zero results in undefined behavior.

In two's complement arithmetic, attempts to divide the smallest signed integer by $- 1$ are attended by similar problems, and are handled with the same range of solutions, from explicit error conditions to undefined behavior.

Most calculators will either return an error or state that 1/0 is undefined, however some TI graphing calculators will evaluate 1/0² to ∞.

[编辑] Historical accidents

On September 21, 1997, a divide by zero error in the USS Yorktown (CG-48) Remote Data Base Manager brought down all the machines on the network, causing the ship's propulsion system to fail. ^[2]

[编辑] 注釋

^ Zero
^ zh-hans:“;zh-hant:「Sunk by Windows NTzh-hans:”;zh-hant:」，Wired News，1998-07-24．

[编辑] 參考

Patrick Suppes 1957 (1999 Dover edition), Introduction to Logic, Dover Publications, Inc., Mineola, New York. ISBN 0-486-40687-3 (pbk.). This book is in print and readily available. Suppes's §8.5 The Problem of Division by Zero begins this way: "That everything is not for the best in this best of all possible worlds, even in mathematics,is well illustrated by the vexing problem of defining the operation of division in the elementary theory of artihmetic" (p. 163). In his §8.7 Five Approaches to Division by Zero he remarks that "...there is no uniformly satisfactory solution" (p. 166)

Charles Seife 2000, Zero: The Biography of a Dangerous Idea, Penguin Books, NY, ISBN 0 14 02.9647 6 (pbk.). This award-winning book is very accessible. Along with the fascinating history of (for some) an abhorent notion and others a cultural asset, describes how zero is misapplied with respect to multiplication and division.

Alfred Tarski 1941 (1995 Dover edition), Introduction to Logic and to the Methodology of Deductive Sciences, Dover Publications, Inc., Mineola, New York. ISBN 0-486-28462-X (pbk.). Tarski's §53 Definitions whose definiendum contains the identity sign discusses how mistakes are made (at least with respect to zero). He ends his chapter "(A discussion of this rather difficult problem [exactly one number satisfying a definiens] will be omitted here.*)" (p. 183). The * points to Exercise #24 (p. 189) wherein he asks for a proof of the following: "In section 53, the definition of the number '0' was stated by way of an example. In order to be certain that this definition does not lead to a contradiction, it should be preceded by the following theorem: There exists exactly one number x such that, for any number y, we have: y + x = y."

[编辑] 延伸閱讀

Jakub Czajko (July 2004) "On Cantorian spacetime over number systems with division by zero ", Chaos, Solitons and Fractals, volume 21, number 2, pages 261–271.

维基新闻相关報導：
British computer scientist's new "nullity" idea provokes reaction from mathematicians

Ben Goldacre（2006年12月7日）．Maths Professor Divides By Zero, Says BBC．

[编辑] 參閱

分类: 零 | 分數 | 数学分析 | 算术

2个隐藏分类: 自2008年4月正在翻譯的條目 | 正在翻譯的條目

See also ebooksgratis.com: no banners, no cookies, totally FREE.

除以零