−2

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
数字-2
名称-2
小写负二
大写负贰
序数词英语Ordinal numeral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Counting rod v-2.png
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

  • 水星在地球上观测的视星等平均约为负二等[75],最大亮度则为−2.48等。[76]
  • 时区UTC-2表示比协调世界时慢2小时[77]
  • 二(三氟甲基)硒维基数据所列Q82391574((CF3)2Se)的沸点为−2 °C。[78]

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节,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]

参考文献

  1. ^ Catherine V. Jeremko. Just in time math (PDF). LearningExpress, LLC, New York. 2003: 20 [2020-03-26]. ISBN 1-57685-506-6. (原始内容存档 (PDF)于2020-03-26). 
  2. ^ Runesson Kempe, Ulla, Anna Lövström, and Björn Hellquist. Beyond the borders of the local: How “instructional products” from learning study can be shared and enhance student learning. International Journal for Lesson and Learning Studies (Emerald Group Publishing Limited). 2018, 7 (2): 111––123. 
  3. ^ Rick Billstein, Shlomo Libeskind, and Johnny W. Lott. A Problem Solving Approach to Mathematics for Elementary School Teachers. Pearson Education, Inc. 2010: 250. 
  4. ^ Sloane, N.J.A. (编). Sequence A061019 (Negate primes in factorization of n.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  5. ^ Can negative numbers be prime?. primes.utm.edu. [2020-03-14]. (原始内容存档于2018-01-23). 
  6. ^ 6.0 6.1 International Bureau of Weights and Measures, The International System of Units (SI) (PDF) 8th, 2006, ISBN 92-822-2213-6 (英语) 
  7. ^ Knuth, Donald. The Art of Computer Programming, Volume 1: Fundamental Algorithms (second edition). Addison-Wesley. 1973: 213–214, also p. 631. ISBN 0-201-03809-9. (原始内容存档于2019-04-03). 
  8. ^ Yan, Michael and Leung, Eric and Han, Binna, The Joy Of Engineering (PDF), 2011-12 [2020-03-21], (原始内容存档 (PDF)于2020-03-21) 
  9. ^ Sloane, N.J.A. (编). Sequence A164793 (smallest number which has in its English name the letter "i" in the n-th position, -1 if such number no exist, -2 for infinite). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  10. ^ Sloane, N.J.A. (编). Sequence A164805. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  11. ^ Horwitz, Kenneth. Extending Fraction Placement from Segments to a Number Line. Children’s Reasoning While Building Fraction Ideas (Springer). 2017: 193––200. 
  12. ^ Haag, VH; et al, Introduction to Algebra (Part 2), ERIC, 1960 
  13. ^ aillon, L and Poon, Chi-Sun and Chiang, YH. Quantifying the waste reduction potential of using prefabrication in building construction in Hong Kong. Waste management (Elsevier). 2009, 29 (1): 309––320. 
  14. ^ Sloane, N.J.A. (编). Sequence A002171 (Glaisher's chi numbers). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  15. ^ Weisstein, Eric W. (编). Complementary Bell Number. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2020-03-12] (英语). 
  16. ^ Amdeberhan, Tewodros and De Angelis, Valerio and Moll, Victor H. Complementary Bell numbers: Arithmetical properties and Wilf’s conjecture. Advances in Combinatorics (Springer). 2013: 23––56. 
  17. ^ 17.0 17.1 17.2 Sloane, N.J.A. (编). Sequence A328933 (Zombie Numbers). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  18. ^ Sloane, N.J.A. (编). Sequence A088306 (Integers n with tan n > |n|). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  19. ^ Hardy, Godfrey Harold; Wright, E. M., An introduction to the theory of numbers Fifth, The Clarendon Press Oxford University Press: 213, 1979 [1938], ISBN 978-0-19-853171-5, MR 0568909 
  20. ^ Conway, John Horton; Guy, Richard K. The Book of Numbers. Springer. 1996: 224. ISBN 0-387-97993-X. 
  21. ^ H.M. Stark. On the “gap” in a theorem of Heegner. Journal of Number Theory. 1969-01, 1 (1): 16–27 [2020-06-19]. doi:10.1016/0022-314X(69)90023-7. (原始内容存档于2020-06-28) (英语). 
  22. ^ Weisstein, Eric W. (编). Heegner Number. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2020-03-14] (英语). 
  23. ^ Sloane, N.J.A. (编). Sequence A003173 (Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  24. ^ Kyle Bradford and Eugen J. Ionascu, Unit Fractions in Norm-Euclidean Rings of Integers, arXiv, 2014 [2020-03-26], (原始内容存档于2020-03-26) 
  25. ^ LeVeque, William J. Topics in Number Theory, Volumes I and II. New York: Dover Publications. 2002: II:57,81 [1956]. ISBN 978-0-486-42539-9. Zbl 1009.11001. 
  26. ^ Kelly Emmrich and Clark Lyons. Norm-Euclidean Ideals in Galois Cubic Fields (PDF). 2017 West Coast Number Theory Conference. 2017-12-18 [2020-03-26]. (原始内容存档 (PDF)于2020-03-26). 
  27. ^ Sloane, N.J.A. (编). Sequence A296818 (Squarefree values of n for which the quadratic field Q[ sqrt(n) ] possesses a norm-Euclidean ideal class.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  28. ^ 28.0 28.1 Sloane, N.J.A. (编). Sequence A229686 (The negative number of minimum absolute value not obtainable from 1 in n steps using addition, multiplication, and subtraction.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  29. ^ Koiran, Pascal. Valiant’s model and the cost of computing integers. computational complexity (Springer). 2005, 13 (3-4): 131––146. 
  30. ^ Shub, Michael and Smale, Steve. On the intractability of Hilbert’s Nullstellensatz and an algebraic version of “NP= P?”. Duke Mathematical Journal. 1995, 81 (1): pp. 47-54. 
  31. ^ Sloane, N.J.A. (编). Sequence A067994 (Hermite numbers). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  32. ^ pahio. Hermite numbers. planetmath.org. 2013-03-22 [2020-03-14]. (原始内容存档于2015-09-19). 
  33. ^ Weisstein, Eric W. (编). Hermite Number. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2020-03-12] (英语). 
  34. ^ Sloane, N.J.A. (编). Sequence A005008. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  35. ^ Sloane, N.J.A. (编). Sequence A123642. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  36. ^ Richard K. Guy. "Figurate Numbers", §D3 in Unsolved Problems in Number Theory,. Problem Books in Mathematics 2nd ed. New York: Springer-Verlag. 1994: 148. ISBN 978-0387208602. 
  37. ^ 37.0 37.1 Sloane, N.J.A. (编). Sequence A165519 (Integers k for which k(k+1)(k+2) is a triangular number.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  38. ^ Sloane, N.J.A. (编). Sequence A214283 (Smallest Euler characteristic of a downset on an n-dimensional cube). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  39. ^ Bard, G.V. Sage for Undergraduates. American Mathematical Society. 2015: 269. ISBN 9781470411114. LCCN 14033572. 
  40. ^ Dresden, Greg; Dymàček, Wayne M. Finding Factors of Factor Rings over the Gaussian Integers. The American Mathematical Monthly. 2005-08-01, 112 (7): 602. doi:10.2307/30037545. 
  41. ^ 41.0 41.1 CHAUNCEY H. WELLS. Using a negative base for number notation. The Mathematics Teacher (National Council of Teachers of Mathematics). 1963, 56 (2): 91––93. ISSN 0025-5769. 
  42. ^ Sloane, N.J.A. (编). Sequence A004171. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  43. ^ Knuth, Donald, The Art of Computer Programming, Volume 2 3rd: 204–205, 1998 . Knuth mentions both negabinary and negadecimal.
  44. ^ The negaternary system is discussed briefly in Marko Petkovsek. Ambiguous Numbers are Dense. The American Mathematical Monthly. 1990-05, 97 (5): 408 [2020-06-19]. doi:10.2307/2324393. (原始内容存档于2020-06-10). 
  45. ^ Sloane, N.J.A. (编). Sequence A122803 (Powers of -2). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  46. ^ Marczynski, R. W., "The First Seven Years of Polish Computing"页面存档备份,存于互联网档案馆), IEEE Annals of the History of Computing, Vol. 2, No 1, January 1980
  47. ^ Robert Braunwart. Negative and Imaginary Radices. School Science and Mathematics. 1965-04, 65 (4): 292–295 [2022-06-23]. doi:10.1111/j.1949-8594.1965.tb13422.x. (原始内容存档于2022-06-27) (英语). 
  48. ^ Leibniz, Gottfried. Probst, S.; Knobloch, E.; Gädeke, N. , 编. Sämtliche Schriften und Briefe, Reihe 7, Band 3: 1672–1676: Differenzen, Folgen, Reihen. Akademie Verlag. 2003: pp.205–207 [2020-03-20]. ISBN 3-05-004003-3. (原始内容存档于2013-10-17). 
  49. ^ Eberhard Knobloch. Beyond Cartesian limits: Leibniz's passage from algebraic to “transcendental” mathematics. Historia Mathematica. 2006-02, 33 (1): 113–131 [2020-06-19]. doi:10.1016/j.hm.2004.02.001. (原始内容存档于2019-10-14) (英语). 
  50. ^ Weisstein, Eric W. (编). Geometric Series. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2020-03-21] (英语). 
  51. ^ Sloane, N.J.A. (编). Sequence A077925. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 
  52. ^ Euler, Leonhard. Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum. 1755: 234 [2020-03-20]. (原始内容存档于2008-02-25). 
  53. ^ Korevaar, Jacob. Tauberian Theory: A Century of Developments. Springer. 2004: 325. ISBN 3-540-21058-X. 
  54. ^ 孙长军. 负二次幂函数与排列数的交错级数型线性微分方程. 山东理工大学学报(自然科学版) (连云港职业技术学院数学教研室). 2004, 05期. 
  55. ^ Alexandre Koyré. An Unpublished Letter of Robert Hooke to Isaac Newton. Isis. 1952-12, 43 (4): 312–337 [2020-06-19]. ISSN 0021-1753. doi:10.1086/348155 (英语). 
  56. ^ Hooke's letter to Newton of 6 Jan. 1680 (Koyré 1952:332)[55]
  57. ^ 中国近代航空工业史(1909-1949). 中国航空工业史丛书: 总史. 航空工业出版社. 2013 [2020-03-22]. ISBN 9787516502617. LCCN 2019437836. (原始内容存档于2020-11-30). 
  58. ^ 58.0 58.1 Havil, J. Gamma: Exploring Euler's Constant. Princeton, New Jersey: Princeton University Press. 2003: 37–42 (Chapter 4). ISBN 0-691-09983-9. 
  59. ^ Evaluating ζ(2) (PDF). secamlocal.ex.ac.uk. [2020-03-21]. (原始内容存档 (PDF)于2007-06-29). 
  60. ^ 许志农. 休閒數學的濫觴⋯中國的洛書 (PDF). lungteng.com.tw. [2020-03-21]. (原始内容存档 (PDF)于2020-03-21). 
  61. ^ 御坂01034. 巴塞尔问题(Basel problem)的多种解法. [2020-03-21]. (原始内容存档于2019-05-02). 
  62. ^ 62.0 62.1 62.2 Knuth, D.E. (1960). "bi-imaginary number system"[63]. Communications of the ACM. 3 (4): 247.
  63. ^ Donald E. Knuth. A imaginary number system. Communications of the ACM. 1960-04-01, 3 (4): 245–247 [2020-06-19]. doi:10.1145/367177.367233. 
  64. ^ 64.0 64.1 Knuth, Donald. Positional Number Systems. The art of computer programming. Volume 2 3rd. Boston: Addison-Wesley. 1998: 205. ISBN 0-201-89684-2. OCLC 48246681. 
  65. ^ 65.0 65.1 Slekys, Arunas G and Avižienis, Algirdas. A modified bi-imaginary number system. 1978 IEEE 4th Symposium onomputer Arithmetic (ARITH) (IEEE). 1978: 48––55. 
  66. ^ 66.0 66.1 Slekys, Arunas George, Design of complex number digital arithmetic units based on a modified bi-imaginary number system., University of California, Los Angeles, 1976 
  67. ^ Kreith, Kurt and Mendle, Al. Toward A Coherent Treatment of Negative Numbers. Journal of Mathematics Education at Teachers College. 2013, 4 (1): 53. 
  68. ^ Walter Noll, Mathematics should not be boring (PDF), CMU Math - Carnegie Mellon University: 13, 2003-03 [2020-03-26], (原始内容存档 (PDF)于2016-03-22) 
  69. ^ Tussy, A.S. and Koenig, D. Prealgebra. Cengage Learning. 2014: 136. ISBN 9781285966052. 
  70. ^ Bofferding, L.C. and Murata, A. and Goldman, S.V. and Okamoto, Y. and Schwartz, D. and Stanford University. School of Education. Expanding the Numerical Central Conceptual Structure: First Graders' Understanding of Integers. Stanford University. 2011: 169. 
  71. ^ 最冷情人節 酷寒襲芝 創77年低溫紀錄. 世界日报. 2020-02-14. 温度降到华氏零下2度 [失效链接]
  72. ^ E.g. Section 4.2.1 in Intel 64 and IA-32 Architectures Software Developer's Manual, Signed integers are two's complement binary values that can be used to represent both positive and negative integer values., Volume 1: Basic Architecture, 2006-11 
  73. ^ 3.9. Two's Complement. Chapter 3. Data Representation. cs.uwm.edu. 2012-12-03 [2014-06-22]. (原始内容存档于2020-11-30). 
  74. ^ David J. Lilja and Sachin S. Sapatnekar, Designing Digital Computer Systems with Verilog, Cambridge University Press, 2005 online页面存档备份,存于互联网档案馆
  75. ^ Abigail Beall. A guide to planet-spotting. New Scientist. 2019-10, 244 (3253): 51 [2020-06-19]. doi:10.1016/S0262-4079(19)32025-1 (英语). 
  76. ^ A. Mallama, J.L. Hilton. Computing apparent planetary magnitudes for The Astronomical Almanac. Astronomy and Computing. 2018-10, 25: 10–24 [2020-06-19]. doi:10.1016/j.ascom.2018.08.002. (原始内容存档于2020-06-15) (英语). 
  77. ^ Current Time Zone. Brazil Considers Having Only One Time Zone. Time and Date. 2009-07-21 [2012-07-14]. (原始内容存档于2012-07-12). 
  78. ^ Macintyre, Jane E. (1994). Dictionary of Inorganic Compounds, Supplement 2页面存档备份,存于互联网档案馆). CRC Press. pp 25.  
  79. ^ 汤佳玲、刘力仁、陈珮伶. 正負2度C數據解讀錯誤 學者不背書. 自由时报. 2010-03-04 [2010-03-06]. (原始内容存档于2010-03-07) (中文(台湾)). 
  80. ^ 朱立群. 環團科學舉證 ±2℃內容有誤. 中国时报. 2010-03-03 [2010-03-06]. (原始内容存档于2014-10-26) (中文(台湾)).