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-2
← −3 −2 −1 →
数表 — 整数

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命名
小写负二
大写负贰
序数词第负二
negative second
识别
种类整数
性质
素因数分解一般不做素因数分解
高斯整数分解<math>i\times \left( 1+i\right) ^{2} </math>
因数1、2
绝对值2
相反数2
表示方式
值-2
算筹File:Counting rod v-2.png
Module:Infobox_number第78行Lua错误：attempt to index field 'wikibase' (a nil value)
二进制−10(2)
三进制−2(3)
四进制−2(4)
五进制−2(5)
八进制−2(8)
十二进制−2(12)
十六进制−2(16)
高斯整数导航
↑
2i
−1+i i 1+i
← −2 −1 0 1 2 →
−1−i −i 1−i
−2i
↓

在数学中，负二是距离原点两个单位的负整数[1]，记作−2或−2[2][3]，是2的加法逆元或相反数，介于−3与−1之间，亦是最大的负偶数。除了少数探讨整环素元的情况外[4]，一般不会将负二视为素数[5]。

负二有时会做为幂次表达平方倒数，用于国际单位制基本单位的表示法中，如m s-2[6]。此外，在部分领域如软件设计，负一通常会作为函数的无效回传值[7]，类似地负二有时也会用于表达除负一外的其他无效情况[8]，例如在整数数列在线大全中，负一作为不存在、负二作为此解是无穷[9][10]。

性质[编辑]

  • 负二为第二大的负整数[11][12]。最大的负整数为负一。因此部分量表会使用负二作为仅次于负一的分数或权重。[13]
  • 负二为负数中最大的偶数，同时也是负数中最大的单偶数（日语：単偶数）。
  • 负二为格莱舍χ数（OEIS数列A002171）[14]
  • 负二为第6个扩展贝尔数[15]（complementary Bell number，或称Rao Uppuluri-Carpenter numbers ）（OEIS数列A000587），前一个是1后一个是-9。[16]
  • 负二为最大的僵尸数[17]，即位数和（首位含负号）的平方与自身的和大于零的负数[17]。前一个为-3（OEIS数列A328933）。所有负数中，只有26个整数有此种性质[17]。
  • 负二为最大能使<math>\tan n > \left| n \right|</math>的负整数[18]。
  • 负二能使二次域<math>\mathbb{Q}[\sqrt{d}]</math>的类数为1，亦即其整数环为唯一分解整环[注 1][19]。而根据史塔克-黑格纳理论（英语：Stark–Heegner theorem），有此性质的负数只有9个[20][21][22]，其对应的自然数称为黑格纳数[23]。
    • 此外负二也能使二次域<math>\mathbb{Q}[\sqrt{d}]</math>成为简单欧几里得整环（simply Euclidean fields，或称欧几里得范数整环，Norm-Euclidean fields）[24]。有此性质的负数只有-11, -7, -3, -2, -1（OEIS数列A048981）[25]。若放宽条件，则负十五也能列入[26][27]。
  • 负二为从1开始使用加法、减法或乘法在2步内无法达到的最大负数[28]。1步内无法达到的最大负数是负一、3步内无法达到的最大负数是负四（OEIS数列A229686）[28]。这个问题为直线问题（英语：straight-line program）与加法、减法和乘法的结合[29]，其通过整数的运算难度对NP = P与否在代数上进行探讨[30]。
  • 负二为2阶的埃尔米特数（英语：Hermite number）[31]，即<math>H_2 = H_2(0) = -2\,</math>[32]。
    • 同时，负二也是唯一一个素的[注 2]埃尔米特数。[33]
  • <math>{ {2!}-{{ {2}^{2} }} } = -2 </math>[34]，同时满足<math>\left| n \right| !-n^2=n</math>，即<math>\left| -2 \right| !-(-2)^2=-2</math>。此外，<math>n!-2^n</math>当<math>n</math>为2和3时结果也为负二[35]。
  • 负二能使k(k+1)(k+2)为三角形数[36]。所有整数只有9个数有此种性质[37]，而负二是有此种性质的最小整数。这9个整数分别为-2, -1, 0, 1, 4, 5, 9, 56和636（OEIS数列A165519）[37]。
  • 负二为立方体下闭集合中欧拉示性数的最小值[38]。

负二的因数[编辑]

负二的拥有的因数若负因数也列入计算则与二的因数（含负因数）相同，为-2、-1、1、2。根据定义一般不对负数进行素因数分解，虽然能将<math>-1</math>提出来[39]计为<math>-1\times 2</math>，因此2可以视为负二的素因数，但不能作为负二的素因数分解结果。虽然不能对负二进行整数分解，由于负二是一个高斯整数，因此可以对负二进行高斯整数分解，结果为<math>i\times(1+i)^2</math>，其中<math>1+i</math>为高斯素数[40]、<math>i</math>为虚数单位。

负二的幂[编辑]

负二的幂<math>(-2)^x</math>示意图

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一个可以代表负二的幂<math>(-2)^x</math>主值的图形，蓝色是实数部、橘色是虚数部、横轴为<math>x</math>、纵轴为<math>(-2)^x</math>。只有在<math>x</math>为整数时<math>(-2)^x</math>为实数

负二的前几次幂为 -2、4、-8、16、-32、64、-128 （OEIS数列A122803）正负震荡[41]，其中正的部分为四的幂、负的部分与四的幂差负二倍[42]，因此这种特性使得负二成为作为底数可以不使用负号、补码等辅助方式表示全体实数的最大负数[41][43][44][45]，并在1957年间有部分计算机采用负二为底之进位制的数字运算进行设计[46]，类似地，使用2i则能表达复数[47]。

负二的幂之和是一个发散几何级数。虽然其结果发散，但仍可以求得其广义之和，其值为1/3[48][49]。

<math>\sum_{k=0}^{n} (-2)^k</math> = 1 − 2 + 4 − 8 + …

若考虑几何级数的计算公式，则有[50]：

<math>\sum_{k=0}^\infty a r^k = \frac{a}{1-r}.</math>

在首项a = 1且公比r = −2时，上述公式的结果为1/3。然而这个级数应为发散级数，其前几项的和为[51]：

1, -1, 3, -5, 11, -21, 43, -85, 171, -341....（OEIS数列A077925）

这个级数虽然发散，然而欧拉对这个级数的结果给出了一个值，即1/3[52]，而这个和称为欧拉之和（英语：Euler summation）[53]。

负二次幂[编辑]

数的负二次幂<math>x^{-2}</math>示意图

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一个可以代表数的负二次幂<math>x^{-2}</math>的函数图形。数的负二次幂亦可以用平方倒数来表示，即<math>x^{-2}=\frac{1}{x^2}</math>

若一数的幂为负二次，则其可以视为平方的倒数，这个部分用于函数也适用[54]，而日常生活中偶尔会用于表示不带除号的单位，如加速度一般计为m/s2，而在国际单位制基本单位的表示法中也可以计为 m s-2[6]。

而平方倒数中较常讨论的议题包括对任意实数<math>n</math>而言，其平方倒数<math>n^{-2}</math>结果恒正、平方反比定律[56]、网格湍流衰减以及巴塞尔问题[57][58]。其中巴塞尔问题指的是自然数的负二次方和（平方倒数和）会收敛并趋近于<math display="inline">\frac{\pi^2} { 6 }</math>，即[59][58]：

<math>\sum^{\infin}_{n=1} {n^{-2}} = {1^{-2} }+{2^{-2} }+{3^{-2} }+\cdots = {1 \over 1^2 }+{1 \over 2^2 }+{1 \over 3^2 }+\cdots = { \pi^2 \over 6 }</math>

而这个值与黎曼ζ函数代入2的结果相同[60][61]。

对任意实数而言，平方倒数的结果恒正。例如负二的平方倒数为四分之一。前几个自然数的平方倒数为：

平方倒数 1 2 3 4 5 6 7 8 9 10
<math>x^{-2}</math> 1 <math>\frac{1}{4}</math> <math>\frac{1}{9}</math> <math>\frac{1}{16}</math> <math>\frac{1}{25}</math> <math>\frac{1}{36}</math> <math>\frac{1}{49}</math> <math>\frac{1}{64}</math> <math>\frac{1}{81}</math> <math>\frac{1}{100}</math>
1 0.25 <math>0.\overline{1}</math> 0.0625 0.04 <math>0.0\overline{27}</math> 0.0204081632....[注 3] 0.015625 <math>0.0\overline{1}2345679\overline{0}</math> 0.01

负二的平方根[编辑]

负二的平方根在定义虚数单位<math>i</math>满足<math>{ {i}^{2} } = -1 </math>后可通过等式<math>\sqrt{-x} = \pm i \sqrt{x}</math>得出，而对负二而言，则为<math>\sqrt{-2} = \pm i \sqrt{2}</math>[注 4][62][64][65][66]。而负二平方根的主值为<math>i \sqrt{2}</math>[注 5]。

表示方法[编辑]

负二通常以在2前方加入负号表示[67]，通常称为“负二”或大写“负贰”，但不应读作“减二”[68]，而在某些场合中，会以“零下二”[69][70]表达-2，例如在表达温度时[71]。

在二进制时，尤其是计算机运算，负数的表示通常会以补码来表示[72]，即将所有位数填上1，再向下减。此时，负二计为“......11111110(2)”，更具体的，4位整数负二计为“1110(2)”；8位整数负二计为“11111110(2)”；16位整数负二计为“1111111111111110(2)”[73]而在使用负号的表示法中，负二计为“-10(2)”[74]。

在其他领域中[编辑]

正负二[编辑]

正负二（<math>\pm 2</math>）是通过正负号表达正二与负二的方式，其可以用来表示4的平方根或二次方程<math>x^2 = 4</math>的解，即<math>\sqrt{4} = \pm{2}</math>。正负二比负二更常出现于文化中，例如一些音乐创作[79]或者纪录片《±2℃》讲述全球气温提升或降低两度对环境可能造成的影响[80][81]。

参见[编辑]

注释[编辑]

  1. ^ 当d<0时，若<math>\mathbb{Q}[\sqrt{d}]</math>的整数环为唯一分解整环，就表示<math>\mathbb{Q}[\sqrt{d}]</math>的数字都只有一种因数分解方式，例如<math>\mathbb{Q}[\sqrt{-5}]</math>的整数环不是唯一分解整环，因为6可以以两种方式在 <math>\mathbb{Z}[\sqrt{-5}]</math> 中表成整数乘积：<math>2\times 3</math> 和 <math>(1+\sqrt{-5})(1-\sqrt{-5})</math>。
  2. ^ 此指埃尔米特多项式的费马伪素数
  3. ^ 7的平方倒数之循环节有42位，0.0204081632 6530612244 8979591836 7346938775 51 ... 参阅49的倒数
  4. ^ 4.0 4.1 bi-imaginary number system<math>\left\langle\sqrt{R},Z_R\right\rangle</math>中，<math>R</math>为负二、<math>Z_R</math>为二的情况<math>\left\langle\pm \mathrm i \sqrt{2},Z_2\right\rangle</math>[62]
  5. ^ 平方根的主值即<math>\sqrt{-x} = \pm i \sqrt{x}</math>取正的值，对于负二而言，即<math>\sqrt{-2} = i \sqrt{2}</math>[注 4][62][64][65][66]

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