2i

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2i
2i
數表高斯整數
<< −3i −2i −i 0  i  2i  3i >>

高斯平面上的位置
命名
數字2i
名稱2i
性質
高斯整數分解
表示方式

代數形式
导航
2i
-1+i i 1+i
-2 -1 0 1 2
-1-i -i 1-i
-2i

是距離原點兩個單位的高斯整數[2]:13,為虛數單位的兩倍[2]:12,同時也是負四的平方根[2]:12[3][4]:ix[5][6][7][8],是方程式的正虛根[3][9]:10。日常生活中通常不會用來計量事物,例如無法具體地描述何謂個人,邏輯上個人並沒有意義。[10]部分書籍或教科書偶爾會使用來做虛數的例子或題目。[11]

高斯平面上,與相鄰的純虛數之高斯整數有,然而複數不具備有序性,即無法判斷間的大小關係,因此無法定義何者為的前一個虛數、何者為的下一個虛數。

性質

  • 不屬於實數,是一個純虛數,同時也是複數位於複數平面,其實部為0、虛部為2[12]輻角為90度(弧度)[13],其也能表達為[14]:7[15]
  • 是一個高斯整數[16][17][18]高斯整數分解[19]:711,或[20]:433,其中,1+i2i的高斯質因數。[19]:711[21][22]:247
  • 所有複數的可以表達為之冪的線性組合[23]換句話說若進位制為底數,則可獨一無二地表示全體複數[24]。該進制稱為2i進制,由高德納1955年發現。[25]
  • 複數的虛數部可以定義為複數與其共軛複數之差除以[26]換言之,則[2]:32
  • 正弦函數可以定義為純虛指數函數與其倒數之差除以的商。[27][28]:41[2]:64
  • 等於最小的質數虛數單位,即[15],其中為第個質數。
  • 虛數單位虛數單位倒數相差
  • 任意數與相乘的意義為模放大兩倍並在複平面上以原點為中心逆時針旋轉90度。[14]:7[2]:20-21

2i的冪

的前幾次冪為1、 2i、 −4、 −8i、 16、 32i、 −64...[29],其會在實部和虛部交錯變換,其單位會在1、i、−1、−i中變化。其中,實數項為−4的冪[30] ,虛數的正值項為16的冪的2倍[31] 、虛數的負值項為16的冪的−8倍[32],因此這種特性使得作為底數可以不將複數表達為實部與虛部就能表示全體複數,[29]並且有研究以此特性設計複數運算電路[33]

2i的平方根

平方根正好是實數單位虛數單位,即[28]:3,反過來說正好是實數單位虛數單位相加的平方,[34][35]:388

相關數字

−2i

的相反數,其平方根曾提議作為複數進位制的底數。[36]

1+i

平方根[28],由於其冪次為1+i、 2i、 −2+2i、 −4、 −4−4i、 −8i...,在正負、虛實交替變化,因此若作為進位制底數可以表達全體複數。但其組合變化相較於為底數的進位制,做為底數更為適合。[37]

−1+i

−1+i進位制系統中整數部分全為零的複數

的平方根。由於其冪次為−1+i、 −2i、 2+2i、 −4、 4−4i、 8i...,其在正負、虛實交替變化,因此其可以構建一個以為底數並用1和0表達複數的進位制[36][38]。其他複數雖然也可以,如,但對高斯整數而言,以為底並不是一個優良的選擇。[37]

除了外,其他形式的複數也能作為進位制底數,但其在表達數的範圍不同,以為例,其表達的範圍較為均勻,而等則會越來越狹長。[39]

參見

參考文獻

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  4. Hart, P. The Book of Imaginary Indians: Ancient Traditions and Modern Caricatures in the White Man's Quest for Meaning. iUniverse. 2008. ISBN 9780595435036. 
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  7. Neuman, Yrsa. Moore’s Paradox and Limits in Language Use. Wittgenstein and the Limits of Language (Routledge). 2019: 159–171. 
  8. Parker, Barry. Fractals. Chaos in the Cosmos (Springer). 1996: 129–154. 
  9. Complex Numbers and the Complex Exponential (PDF). people.math.wisc.edu. [2022-06-23]. 
  10. Abubakr, Mohammed. On logical extension of algebraic division. arXiv preprint arXiv:1101.2798. 2011 [2022-06-23]. doi:10.48550/ARXIV.1101.2798. 
  11. 中學數學實用詞典, 九章出版社, 孫文先, P.22 中的示範其解為2i, ISBN 957-603-093-5
  12. Complex Numbers 2i (PDF). ichthyosapiens.com. [2022-06-23]. 
  13. All Complex Number Solutions z=2i. mathway.com. [2022-06-23]. 
  14. 14.0 14.1 Jeremy Orloff. Complex algebra and the complex plane (PDF). math.mit.edu. [2022-06-23]. 
  15. 15.0 15.1 Wolfram, Stephen. "(smallest prime number) * (imaginary unit)". from Wolfram Alpha: Computational Knowledge Engine, Wolfram Research (英语). 
  16. C. F. Gauss, Theoria residuorum biquadraticorum. Commentatio secunda., Comm. Soc. Reg. Sci. Gottingen 7 (1832) 1-­34; reprinted in Werke, Georg Olms Verlag, Hildesheim, 1973, pp. 93-­148.
  17. 从数到环:环论的早期历史 ,由Israel Kleiner所作 (Elem. Math. 53 (1998) 18 – 35)
  18. Ribenboim, Paulo, The New Book of Prime Number Records, New York: Springer, 1996, ISBN 0-387-94457-5 
  19. 19.0 19.1 Banerjee, Ashmi and Mukherjee, Shaunak and Datta, Somjit and Majumder, Subhashis. Computational search for Gaussian perfect integers. 2015 International Conference on Control Communication & Computing India (ICCC) (IEEE). 2015: 710–715 [2022-08-15]. 
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