0的奇偶性

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0是一个偶数。来证明0是偶数的最简单的方法是检查0是否符合偶数的定义:若某数是2的整数倍数,那么它就是偶数。因为0=0×2,所以0为偶数。除此以外,0还满足偶数的所有性质:0可以被2整除;与0相邻的两个数字都是奇数;0可以被等分成两份。

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,

为什么0是偶数[编辑]

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。[2]

基础解释[编辑]

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[3]

数字是用来计数的,人们用一个数字来表示集合元素的个数。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。[4]

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。[5]

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。

EvenOddNumberLine.svg

一个偶数之后的第二位数字是偶数,没有任何理由跳过0[6]

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。[7]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。[8]

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。[9]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。[10]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。

数学背景[编辑]

参考[编辑]

  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. ^ Ball, Lewis & Thames (2008, p. 15) 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.
  3. ^ Compare Lichtenberg (1972, p. 535) Fig. 1
  4. ^ 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."
  5. ^ Dickerson & Pitman 2012, p. 191.
  6. ^ 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."
  7. ^ Caldwell & Xiong 2012, pp. 5–6.
  8. ^ 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).
  9. ^ Partee 1978,第xxi页
  10. ^ Stewart 2001,第54页 These rules are given, but they are not quoted verbatim.

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