−2
在数学中,负二是距离原点两个单位的负整数[1],记作−2[2]或−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]。
| ||||
---|---|---|---|---|
| ||||
命名 | ||||
数字 | -2 | |||
名称 | -2 | |||
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 | ||||
值 | -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) | |||
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。
- 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 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,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]。
- 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[31],即 [32]。
- [34],同时满足 ,即 。此外, 当 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 为高斯素数[40]、 为虚数单位。
负二的幂
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]。
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]:
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[53]。
负二次幂
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]:
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
- 2
- 2i
注释
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