−2

最大的負偶數
(重定向自平方倒數

數學中,負二是距離原點兩個單位的負整數[1],记作−2[2]2[3],是2加法逆元相反數,介於−3−1之間,亦是最大的負偶數。除了少數探討整環質元素的情況外[4],一般不會將負二視為質數[5]

-2
← −3 −2 −1 →
数表整数

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

命名
小寫負二
大寫負貳
序數詞第負二
negative second
識別
種類整數
性質
質因數分解一般不做質因數分解
高斯整數分解
因數1、2
絕對值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)
高斯整數導航
2i
−1+i i 1+i
−2 −1 0 1 2
−1−i i 1−i
−2i

負二有時會做為冪次表達平方倒數,用於國際單位制基本單位的表示法中,如m s-2[6]。此外,在部份領域如軟體設計負一通常會作為函數的無效回傳值[7],類似地負二有時也會用於表達除負一外的其他無效情況[8],例如在整數數列線上大全中,負一作為不存在、負二作為此解是无穷[9][10]

性質

  • 負二為第二大的負整數[11][12]。最大的負整數為負一。因此部分量表會使用負二作為僅次於負一的分數或權重。[13]
  • 負二為負數中最大的偶數,同時也是負數中最大的單偶數日语単偶数
  • 負二為格萊舍χ數(OEIS數列A002171[14]
  • 負二為第6個擴充貝爾數[15](complementary Bell number,或稱Rao Uppuluri-Carpenter numbers )(OEIS數列A000587),前一個是1後一個是-9。[16]
  • 負二為最大的殭屍數[17],即位數和(首位含負號)的平方與自身的和大於零的負數[17]。前一個為-3(OEIS數列A328933)。所有負數中,只有26個整數有此種性質[17]
  • 負二為最大能使 的負整數[18]
  • 負二能使二次域 類数為1,亦即其整數環唯一分解整環[註 1][19]。而根據史塔克-黑格纳理論英语Stark–Heegner theorem,有此性質的負數只有9個[20][21][22],其對應的自然數稱為黑格纳数[23]
    • 此外負二也能使二次域 成為簡單歐幾里得整環(simply Euclidean fields,或稱歐幾里得範數整環,Norm-Euclidean fields)[24]。有此性質的負數只有-11, -7, -3, -2, -1(OEIS數列A048981[25]。若放寬條件,則負十五也能列入[26][27]
  • 負二為從1開始使用加法、減法或乘法在2步內無法達到的最大負數[28]。1步內無法達到的最大負數是負一、3步內無法達到的最大負數是負四(OEIS數列A229686[28]。這個問題為直線問題英语straight-line program與加法、減法和乘法的結合[29],其透過整數的運算難度對NP = P與否在代數上進行探討[30]
  • 負二為2階的埃尔米特数英语Hermite number[31],即 [32]
    • 同時,負二也是唯一一個素的[註 2]埃尔米特数。[33]
  •  [34],同時滿足 ,即 。此外,  為2和3時結果也為負二[35]
  • 負二能使k(k+1)(k+2)為三角形數[36]。所有整數只有9個數有此種性質[37],而負二是有此種性質的最小整數。這9個整數分別為-2, -1, 0, 1, 4, 5, 9, 56和636(OEIS數列A165519[37]
  • 負二為立方體下闭集合欧拉示性数的最小值[38]

負二的因數

負二的擁有的因數若負因數也列入計算則與二的因數(含負因數)相同,為-2、-1、1、2。根據定義一般不對負數進行質因數分解,雖然能將 提出來[39]計為 ,因此2可以視為負二的質因數,但不能作為負二的質因數分解結果。雖然不能對負二進行整數分解,由於負二是一個高斯整數,因此可以對負二進行高斯整數分解,結果為 ,其中 高斯質數[40] 虛數單位

負二的冪

負二的冪 示意圖
一個可以代表負二的冪 主值的圖形,藍色是實數部、橘色是虛數部、橫軸為 、縱軸為 。只有在 為整數時 為實數

負二的前幾次冪為 -2、4、-8、16、-32、64、-128 (OEIS數列A122803)正負震盪[41],其中正的部分為四的冪、負的部分與四的冪差負二倍[42],因此這種特性使得負二成為作為底數可以不使用負號、二補數等輔助方式表示全體實數的最大負數[41][43][44][45],並在1957年間有部分計算機採用負二為底之進位制的數字運算進行設計[46],類似地,使用2i則能表達複數[47]

負二的冪之和是一個发散几何级数。雖然其結果發散,但仍可以求得其廣義之和,其值為1/3[48][49]

  = 1 − 2 + 4 − 8 + …

若考慮几何级数的計算公式,則有[50]

 

在首項a = 1且公比r = −2時,上述公式的結果為1/3。然而這個級數應為發散級數,其前幾項的和為[51]

1, -1, 3, -5, 11, -21, 43, -85, 171, -341....(OEIS數列A077925

這個級數雖然發散,然而歐拉對這個級數的結果給出了一個值,即1/3[52],而這個和稱為歐拉之和英语Euler summation[53]

負二次冪

數的負二次冪 示意圖
一個可以代表數的負二次冪 函數圖形。數的負二次冪亦可以用平方倒數來表示,即 

若一數的冪為負二次,則其可以視為平方的倒數,這個部分用於函數也適用[54],而日常生活中偶爾會用于表示不帶除號的單位,如加速度一般計為m/s2,而在國際單位制基本單位的表示法中也可以計為 m s-2[6]

而平方倒數中較常討論的議題包括對任意實數 而言,其平方倒數 結果恆正、平方反比定律[56]、网格湍流衰減[57]以及巴塞尔问题[58]。其中巴塞尔问题指的是自然數的負二次方和(平方倒數和)會收斂並趨近於 ,即[59][58]

 

而這個值與黎曼ζ函數代入2的結果相同[60][61]

對任意實數而言,平方倒數的結果恆正。例如負二的平方倒數為四分之一。前幾個自然數的平方倒數為:

平方倒數 1 2 3 4 5 6 7 8 9 10
  1                  
1 0.25   0.0625 0.04   0.0204081632....[註 3] 0.015625   0.01

負二的平方根

負二的平方根在定義虛數單位 滿足 後可透過等式 得出,而對負二而言,則為 [註 4][62][64][65][66]。而負二平方根的主值為 [註 5]

表示方法

負二通常以在2前方加入負號表示[67],通常稱為「負二」或大寫「負貳」,但不應讀作「減二」[68],而在某些場合中,會以「零下二」[69][70]表達-2,例如在表達溫度時[71]

在二進制時,尤其是計算機運算,負數的表示通常會以二補數來表示[72],即將所有位數填上1,再向下減。此時,負二計為「......11111110(2)」,更具體的,4位元整數負二計為「1110(2)」;8位元整數負二計為「11111110(2)」;16位元整數負二計為「1111111111111110(2)[73]而在使用負號的表示法中,負二計為「-10(2)[74]

在其他領域中

正負二

正負二( )是透過正負號表達正二與負二的方式,其可以用來表示4的平方根或二次方程 的解,即 。正負二比負二更常出現於文化中,例如一些音樂創作[79]或者紀錄片《±2℃》講述全球氣溫提升或降低兩度對環境可能造成的影響[80][81]

參見

註釋

  1. ^ 當d<0時,若 的整數環為唯一分解整環,就表示 的數字都只有一種因數分解方式,例如 的整數環不是唯一分解整環,因為6可以以兩種方式在   中表成整數乘積:  
  2. ^ 此指埃尔米特多项式费马伪素数
  3. ^ 7的平方倒數之循環節有42位,0.0204081632 6530612244 8979591836 7346938775 51 ... 參閱49的倒數
  4. ^ 4.0 4.1 bi-imaginary number system 中, 為負二、 為二的情況 [62]
  5. ^ 平方根的主值即 取正的值,對於負二而言,即 [註 4][62][64][65][66]

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