−2

最大的負偶數

数学中,负二是距离原点两个单位的负整数[1],记作−2[2]2[3],是2加法逆元相反数,介于−3−1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]

-2
← −3 −2 −1 →
数表整数

<<  −10  −9‍  −8‍ −7 −6  −5‍ −4 −3 −2 −1 >>

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
约数1、2
绝对值2
相反数2
表示方式
-2
算筹
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

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

  • 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 的负整数[18]
  • 负二能使二次域 类数为1,亦即其整数环唯一分解整环[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 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],其透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 [32]
    • 同时,负二也是唯一一个素的[注 2]埃尔米特数。[33]
  •  [34],同时满足 ,即 。此外,  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欧拉示性数的最小值[38]

负二的约数

负二的拥有的约数若负约数也列入计算则与二的约数(含负约数)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 提出来[39]计为 ,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 ,其中 高斯素数[40] 虚数单位

负二的幂

负二的幂 示意图
一个可以代表负二的幂 主值的图形,蓝色是实数部、橘色是虚数部、横轴为 、纵轴为 。只有在 为整数时 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]

  = 1 − 2 + 4 − 8 + …

若考虑几何级数的计算公式,则有[50]

 

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

数的负二次幂 示意图
一个可以代表数的负二次幂 函数图形。数的负二次幂亦可以用平方倒数来表示,即 

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 而言,其平方倒数 结果恒正、平方反比定律[56]、网格湍流衰减[57]以及巴塞尔问题[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 ,即[59][58]

 

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
  1                  
1 0.25   0.0625 0.04   0.0204081632....[注 3] 0.015625   0.01

负二的平方根

负二的平方根在定义虚数单位 满足 后可透过等式 得出,而对负二而言,则为 [注 4][62][64][65][66]。而负二平方根的主值为 [注 5]

表示方法

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

正负二

正负二( )是透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 的解,即 。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

注释

  1. ^ 当d<0时,若 的整数环为唯一分解整环,就表示 的数字都只有一种约数分解方式,例如 的整数环不是唯一分解整环,因为6可以以两种方式在   中表成整数乘积:  
  2. ^ 此指埃尔米特多项式费马伪素数
  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 中, 为负二、 为二的情况 [62]
  5. ^ 平方根的主值即 取正的值,对于负二而言,即 [注 4][62][64][65][66]

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