−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

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 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

编辑

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

编辑
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

编辑
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

编辑

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

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( )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

编辑

注释

编辑
  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]

参考文献

编辑
  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. ISBN 9780412491009.
  79. ^ Pace, Ian. Positive or negative 2. The Musical Times (JSTOR). 1998, 139 (1860): 4––15. 
  80. ^ 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)). 
  81. ^ 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)).