0的奇偶性

0是一个偶数。按照定义，若某数是2的整数倍数，那么它就是偶数，而0＝0×2，所以0为偶数。^[1]

0还满足其它一些由偶数构建出来的一些模型，例如在算术运算中的一些奇偶规则：偶数－偶数＝偶数。

数学背景

一千多年以来，数学家一直难以解决数字零，非数学家仍不确定如何将其分类。巴比伦人和古希腊人使用它来区分大小，例如：26 和 206。在此之前，人们只能根据上下文的使用，来判断一个数字是否大于另一个数字。13世纪，意大利数学家斐波那契（Fibonacci）是第一个在欧洲普及阿拉伯数字的人。他将数字一到九分类为数字，而将零分类为符号。^[2]

根据剑桥大学千年数学项目詹姆斯·格莱姆博士的说法：“1990年代的反应时间实验表明，人们在决定零是奇数还是偶数时要慢10％。”孩子们发现很难识别零是奇数还是偶数，“1990年代对小学生的一项调查显示，约50％认为零是偶数，约20％认为零是奇数，其余30％认为两者都不是”格莱姆表示：“直到1600年，持续辩论和抗争之后，零才真正被接受为偶数。”^[2]

欧几里得并未将1视为数字，但现在1也视为奇数。^[3]

欧几里得的元素

前几个奇数是3、5、7、9、11。^[4]
前几个偶数是2、4、6、8、10。^[4]

数论中众多结论援引了算术基本定理和偶数的代数性质，因此上述选择具有深远的影响。例如，正数有唯一的整数分解这一事实，意味着我们可以确定一个数有偶数个不同的质因数还是奇数个不同的质因数。因为1不是素数，也没有素数因子，是空积；因为0是偶数，所以1有偶数个不同的质因数。这意味着默比乌斯函数的值μ(1) = 1，这对于积性函数和默比乌斯反演公式是很有必要的。^[5]

为什么0是偶数

《大英百科全书》记载为：“大多数人都对数字0感到困惑，不确定它是否作为整数的起始，并且不知道其作为数字的位置。从技术上讲，它表示空集。”奇偶性（Parity）^[6]是先期数学课程中最早学习的规则^[3]，将所有整数分成两类的方式：偶数和奇数^[7]。偶数的最基础的定义就可以直接用来证明0是偶数。偶数的定义是：如果一个数是2的整数倍数，那么这个数便是偶数。例如：因为10＝5×2，所以10是偶数。同样的，因为0＝0×2，所以0是偶数。^[1]除了使用偶数的定义这样一种证明方式来证明0是一个偶数以外，还有其它的方法来证明0是一个偶数。^[8]

基础解释

On the left, boxes with 0, 2, and 4 white objects in pairs; on the right, 1, 3, and 5 objects, with the unpaired object in red — 有0个元素的集合没有红色元素剩余^[9]

数字是用来计数的，人们用一个数字来表示集合内元素的个数。0则对应这没有元素，即空集中元素的个数。对数分奇偶就是为了将集合中的元素分为两部分。如果一个集合中的元素可两两配对且没有剩余，那么这个集合的基数便是偶数。如果有一个元素剩余，那么这个集合的基数便是奇数。在此定义之下，因为空集可以被分为两份并且没有元素剩余，所以0是一个偶数。^[10]

还有一种更为具象的偶数定义：如果一个集合中元素可以分成基数相同的两个集合，那么这个集合的基数为偶数，否则为奇数。这个定义与上一个定义是等价的。在此定义之下，因为空集可以分成2个基数都为0的集合，所以0是偶数。^[11]

数字可以用数轴来可视化表现，其中有个常见的特征奇数和偶数相互交替。当负数也算入其中时，这个特征变得尤为明显。

一个偶数之后的第二位数字是偶数，没有任何理由跳过0。^[12]

上述的定义使用了一些数学术语，例如偶数可以被2整除，这一定义归根到底是一个约定。和偶数不同，一些数学术语有目的的排除一些平凡或退化的情况。素数是一个非常有名的例子。在20世纪之前，素数的定义是不一致的，包括克里斯蒂安·哥德巴赫、约翰·海因里希·兰伯特、阿德里安-马里·勒让德、阿瑟·凯莱在内的一些非常著名的数学家都曾经在著作中写过0是一个素数。^[13]现在对素数的定义是：如果一个数有且只有1和本身两个约数，那么这个数是素数。因为1只有一个约数，所以1不是一个素数。这个定义因为更加适用于很多有关素数的数学理论而被广泛接受。例如，当1不再被认为是一个素数时，算术基本定理的表述才更加简单，容易。^[14]

既然素数可以并不包括1，那么偶数似乎也可以并不包括0。但是在这种情况下，一些和偶数有关的数学理论变得难以表述，甚至和奇偶数有关的四则运算都要受到影响。例如，奇偶数运算中存在着以下规则：

偶数±偶数＝偶数

奇数±奇数＝偶数

偶数×整数＝偶数

在这些式子的左侧填入适当的数字可以使得右边为0：

2-2=0

-3+3=0

4×0＝0

显而易见地是，这些规则将会因为0不是一个偶数而变得不正确。^[15]不过，一些坚持0不是偶数的人并不会因此改变自己的观点，他们会加上一些特例来保证运算规则的正确性。例如，一个考试指南规定：0既不是偶数也不是奇数。^[16]这样，上述有关奇偶数的运算规则就必须加上一些例外：

偶数±偶数＝偶数（或0）

奇数±奇数＝偶数（或0）

偶数×整数＝偶数（或0）

将0排除在偶数之外使得很多有关偶数的规则、定理都要加上类似的例外。

参考

^ ^1.0 ^1.1 Penner 1999，第34页: Lemma B.2.2, The integer 0 is even and is not odd. Penner uses the mathematical symbol ∃, the existential quantifier, to state the proof: "To see that 0 is even, we must prove that ∃k (0 = 2k), and this follows from the equality 0 = 2 ⋅ 0."
^ ^2.0 ^2.1 Laura Gray. Is zero an even number?. BBC News. 2012-12-02 [2020-02-06]. （原始内容存档于2017-12-28）（英语）.
^ ^3.0 ^3.1 David E. Joyce. 7. An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number.. Department of Mathematics and Computer Science Clark University. 1997 [2020-02-06]. （原始内容存档于2020-02-03）（英语）. On Definition 6: The definition even number is clear: the number a is even if it is of the form b + b. The first few even numbers are 2, 4, 6, 8, 10. On Definition 7: The definition for odd number has two statements. The first can be taken as a definition of odd number, a number which is not divisible into two equal parts, that is to say not an even number. The first few odd numbers are 3, 5, 7, 9, 11. Euclid did not treat 1 as a number, but now 1 is also considered an odd number.
^ ^4.0 ^4.1 David E. Joyce. Definitions 6–7. Department of Mathematics and Computer Science Clark University. 1997 [2020-02-06]. （原始内容存档于2020-02-03）（英语）. The other statement is not a definition for odd number, since one has already been given, but an unproved statement. It is easy to recognize that something has to be proved, since if we make the analogous definitions for another number, say 10, then analogous statement is false. Suppose we say a “decade number” is one divisible by 10, and and “undecade number” is one not divisible by 10. Then it is not the case that an undecade number differs by a unit from a decade number; the number 13, for instance, is not within 1 of a decade number. The unproved statement that a number differing from an even number by 1 is an odd number ought to be proved. That statement is used in proposition IX.22 and several propositions that follow it. It could be proved using, for instance, a principle that any decreasing sequence of numbers is finite.
^ Devlin 1985，第30–33页
^ Nana Ho. 為什麼 0 是偶數？. 科技新报. 2020-02-04 [2020-02-06]. （原始内容存档于2020-02-06）（中文（台湾））.
^ Jonathan Hogeback. Is Zero an Even or an Odd Number?. Encyclopædia Britannica. [2020-02-06]. （原始内容存档于2019-08-11）（英语）. So where exactly does 0 fall into these categories? Most people are confused by the number 0, unsure if it’s an integer to begin with and unaware of its placement as a number, because it technically signifies an empty set. Under the rules of parity, is zero even or odd?
^ Ball, Lewis & Thames (2008，第15页) discuss this challenge for the elementary-grades teacher, who wants to give mathematical reasons for mathematical facts, but whose students neither use the same definition, nor would understand it if it were introduced.
^ Compare Lichtenberg (1972，第535页) Fig. 1
^ Lichtenberg 1972，第535–536页 "...numbers answer the question How many? for the set of objects ... zero is the number property of the empty set ... If the elements of each set are marked off in groups of two ... then the number of that set is an even number."
^ Dickerson & Pitman 2012，第191页.
^ Lichtenberg 1972，第537页; compare her Fig. 3. "If the even numbers are identified in some special way ... there is no reason at all to omit zero from the pattern."
^ Caldwell & Xiong 2012，第5–6页.
^ Gowers 2002，第118页 "The seemingly arbitrary exclusion of 1 from the definition of a prime … does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes." For a more detailed discussion, see Caldwell & Xiong (2012).
^ Partee 1978，第xxi页
^ Stewart 2001，第54页 These rules are given, but they are not quoted verbatim.

外部链接

Doctor Rick, Is Zero Even?, Ask Dr. Math (The Math Forum), 2001 [2013-06-06], （原始内容存档于2013-12-15）
Straight Dope Science Advisory Board, Is zero odd or even?, The Straight Dope Mailbag, 1999 [2013-06-06], （原始内容存档于2012-12-27）
Is Zero Even? - Numberphile（页面存档备份，存于互联网档案馆）, video with Dr. James Grime, University of Nottingham

[#1-1] 1.0 ^1.1 Penner 1999，第34页: Lemma B.2.2, The integer 0 is even and is not odd. Penner uses the mathematical symbol ∃, the existential quantifier, to state the proof: "To see that 0 is even, we must prove that ∃k (0 = 2k), and this follows from the equality 0 = 2 ⋅ 0."

[BBC_News-2] 2.0 ^2.1 Laura Gray. Is zero an even number?. BBC News. 2012-12-02 [2020-02-06]. （原始内容存档于2017-12-28）（英语）.

[Euclid’s_Elements-3] 3.0 ^3.1 David E. Joyce. 7. An odd number is that which is not divisible into two equal parts, or that which differs by a unit from an even number.. Department of Mathematics and Computer Science Clark University. 1997 [2020-02-06]. （原始内容存档于2020-02-03）（英语）. On Definition 6: The definition even number is clear: the number a is even if it is of the form b + b. The first few even numbers are 2, 4, 6, 8, 10. On Definition 7: The definition for odd number has two statements. The first can be taken as a definition of odd number, a number which is not divisible into two equal parts, that is to say not an even number. The first few odd numbers are 3, 5, 7, 9, 11. Euclid did not treat 1 as a number, but now 1 is also considered an odd number.

[歐幾里得的元素-4] 4.0 ^4.1 David E. Joyce. Definitions 6–7. Department of Mathematics and Computer Science Clark University. 1997 [2020-02-06]. （原始内容存档于2020-02-03）（英语）. The other statement is not a definition for odd number, since one has already been given, but an unproved statement. It is easy to recognize that something has to be proved, since if we make the analogous definitions for another number, say 10, then analogous statement is false. Suppose we say a “decade number” is one divisible by 10, and and “undecade number” is one not divisible by 10. Then it is not the case that an undecade number differs by a unit from a decade number; the number 13, for instance, is not within 1 of a decade number. The unproved statement that a number differing from an even number by 1 is an odd number ought to be proved. That statement is used in proposition IX.22 and several propositions that follow it. It could be proved using, for instance, a principle that any decreasing sequence of numbers is finite.

[5] Devlin 1985，第30–33页

[6] Nana Ho. 為什麼 0 是偶數？. 科技新报. 2020-02-04 [2020-02-06]. （原始内容存档于2020-02-06）（中文（台湾））.

[7] Jonathan Hogeback. Is Zero an Even or an Odd Number?. Encyclopædia Britannica. [2020-02-06]. （原始内容存档于2019-08-11）（英语）. So where exactly does 0 fall into these categories? Most people are confused by the number 0, unsure if it’s an integer to begin with and unaware of its placement as a number, because it technically signifies an empty set. Under the rules of parity, is zero even or odd?

[8] Ball, Lewis & Thames (2008，第15页) discuss this challenge for the elementary-grades teacher, who wants to give mathematical reasons for mathematical facts, but whose students neither use the same definition, nor would understand it if it were introduced.

[9] Compare Lichtenberg (1972，第535页) Fig. 1

[10] Lichtenberg 1972，第535–536页 "...numbers answer the question How many? for the set of objects ... zero is the number property of the empty set ... If the elements of each set are marked off in groups of two ... then the number of that set is an even number."

[FOOTNOTEDickersonPitman2012191-11] Dickerson & Pitman 2012，第191页.

[12] Lichtenberg 1972，第537页; compare her Fig. 3. "If the even numbers are identified in some special way ... there is no reason at all to omit zero from the pattern."

[FOOTNOTECaldwellXiong20125–6-13] Caldwell & Xiong 2012，第5–6页.

[14] Gowers 2002，第118页 "The seemingly arbitrary exclusion of 1 from the definition of a prime … does not express some deep fact about numbers: it just happens to be a useful convention, adopted so there is only one way of factorizing any given number into primes." For a more detailed discussion, see Caldwell & Xiong (2012).

[Partee-15] Partee 1978，第xxi页

[MarkAlanStewart-16] Stewart 2001，第54页 These rules are given, but they are not quoted verbatim.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

数学背景

欧几里得的元素

为什么0是偶数

基础解释

参考

相关书籍

外部链接