-2

最大的负偶数
(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
數字-2
名稱-2
小寫負二
大寫負貳
序數詞英语Ordinal numeral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
表示方式
-2
算筹Counting rod v-2.png
二進制-10(2)
八進制-2(8)
十二進制-2(12)
十六進制-2(16)

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]

性質编辑

  • 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中的数列A002171[14]
  • 負二為第6個擴充貝爾數[15](complementary Bell number,或稱Rao Uppuluri-Carpenter numbers )(OEIS中的数列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中的数列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中的数列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中的数列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中的数列A165519[37]
  • 負二為立方體下闭集合欧拉示性数的最小值[38]

負二的因數编辑

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

負二的冪编辑

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中的数列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]

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[47][48]

  = 1 − 2 + 4 − 8 + …

若考慮几何级数的計算公式,則有[49]

 

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[50]

1, -1, 3, -5, 11, -21, 43, -85, 171, -341....(OEIS中的数列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[51],而這個和稱為歐拉之和英语Euler summation[52]

負二次冪编辑

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[53],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[55]、网格湍流衰減[56]以及巴塞尔问题[57]。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[58][57]

 

而這個值與黎曼ζ函數代入2的結果相同[59][60]

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

負二的平方根编辑

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][61][63][64][65]。而負二平方根的主值為 [註 5]

表示方法编辑

負二通常以在2前方加入負號表示[66],通常稱為「負二」或大寫「負貳」,但不應讀作「減二」[67],而在某些場合中,會以「零下二」[68][69]表達-2,例如在表達溫度時[70]

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[71],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)[72]而在使用負號的表示法中,負二計為「-10(2)[73]

在其他領域中编辑

正負二编辑

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[78]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[79][80]

參見编辑

註釋编辑

  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節,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 [61]
  5. ^ 平方根的主值即 取正的值,對於負二而言,即 [註 4][61][63][64][65]

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