−2

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-2
← −3 −2 −1 →
數表 — 整數

<<  −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<math>i\times \left( 1+i\right) ^{2} </math>
因數1、2
絕對值2
相反數2
表示方式
值-2
算籌File:Counting rod v-2.png
Module:Infobox_number第78行Lua錯誤：attempt to index field 'wikibase' (a nil value)
二進制−10(2)
三進制−2(3)
四進制−2(4)
五進制−2(5)
八進制−2(8)
十二進制−2(12)
十六進制−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
↓

在數學中，負二是距離原點兩個單位的負整數[1]，記作−2或−2[2][3]，是2的加法反元素或相反數，介於−3與−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]。

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，在部份領域如軟件設計，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、負二作為此解是無窮[9][10]。

性質[編輯]

  • 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（日語：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<math>\tan n > \left| n \right|</math>的負整數[18]。
  • 負二能使二次體<math>\mathbb{Q}[\sqrt{d}]</math>的類數為1，亦即其整數環為唯一分解整環[註 1][19]。而根據史塔克-黑格納理論（英語：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納數[23]。
    • 此外負二也能使二次體<math>\mathbb{Q}[\sqrt{d}]</math>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（英語：straight-line program）與加法、減法和乘法的結合[29]，其通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階的埃爾米特數（英語：Hermite number）[31]，即<math>H_2 = H_2(0) = -2\,</math>[32]。
    • 同時，負二也是唯一一個素的[註 2]埃爾米特數。[33]
  • <math>{ {2!}-{{ {2}^{2} }} } = -2 </math>[34]，同時滿足<math>\left| n \right| !-n^2=n</math>，即<math>\left| -2 \right| !-(-2)^2=-2</math>。此外，<math>n!-2^n</math>當<math>n</math>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]。
  • 負二為立方體下閉集合中歐拉示性數的最小值[38]。

負二的因數[編輯]

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<math>-1</math>提出來[39]計為<math>-1\times 2</math>，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<math>i\times(1+i)^2</math>，其中<math>1+i</math>為高斯質數[40]、<math>i</math>為虛數單位。

負二的冪[編輯]

負二的冪<math>(-2)^x</math>示意圖

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一個可以代表負二的冪<math>(-2)^x</math>主值的圖形，藍色是實數部、橘色是虛數部、橫軸為<math>x</math>、縱軸為<math>(-2)^x</math>。只有在<math>x</math>為整數時<math>(-2)^x</math>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、二補碼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發散幾何級數。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]。

<math>\sum_{k=0}^{n} (-2)^k</math> = 1 − 2 + 4 − 8 + …

若考慮幾何級數的計算公式，則有[50]：

<math>\sum_{k=0}^\infty a r^k = \frac{a}{1-r}.</math>

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（英語：Euler summation）[53]。

負二次冪[編輯]

數的負二次冪<math>x^{-2}</math>示意圖

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一個可以代表數的負二次冪<math>x^{-2}</math>的函數圖形。數的負二次冪亦可以用平方倒數來表示，即<math>x^{-2}=\frac{1}{x^2}</math>

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於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<math>n</math>而言，其平方倒數<math>n^{-2}</math>結果恆正、平方反比定律[56]、網格湍流衰減以及巴塞爾問題[57][58]。其中巴塞爾問題指的是自然數的負二次方和（平方倒數和）會收斂並趨近於<math display="inline">\frac{\pi^2} { 6 }</math>，即[59][58]：

<math>\sum^{\infin}_{n=1} {n^{-2}} = {1^{-2} }+{2^{-2} }+{3^{-2} }+\cdots = {1 \over 1^2 }+{1 \over 2^2 }+{1 \over 3^2 }+\cdots = { \pi^2 \over 6 }</math>

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
<math>x^{-2}</math> 1 <math>\frac{1}{4}</math> <math>\frac{1}{9}</math> <math>\frac{1}{16}</math> <math>\frac{1}{25}</math> <math>\frac{1}{36}</math> <math>\frac{1}{49}</math> <math>\frac{1}{64}</math> <math>\frac{1}{81}</math> <math>\frac{1}{100}</math>
1 0.25 <math>0.\overline{1}</math> 0.0625 0.04 <math>0.0\overline{27}</math> 0.0204081632....[註 3] 0.015625 <math>0.0\overline{1}2345679\overline{0}</math> 0.01

負二的平方根[編輯]

負二的平方根在定義虛數單位<math>i</math>滿足<math>{ {i}^{2} } = -1 </math>後可通過等式<math>\sqrt{-x} = \pm i \sqrt{x}</math>得出，而對負二而言，則為<math>\sqrt{-2} = \pm i \sqrt{2}</math>[註 4][62][64][65][66]。而負二平方根的主值為<math>i \sqrt{2}</math>[註 5]。

表示方法[編輯]

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碼來表示[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[編輯]

正負二[編輯]

正負二（<math>\pm 2</math>）是通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<math>x^2 = 4</math>的解，即<math>\sqrt{4} = \pm{2}</math>。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[編輯]

註釋[編輯]

  1. ^ 當d<0時，若<math>\mathbb{Q}[\sqrt{d}]</math>的整數環為唯一分解整環，就表示<math>\mathbb{Q}[\sqrt{d}]</math>的數字都只有一種因數分解方式，例如<math>\mathbb{Q}[\sqrt{-5}]</math>的整數環不是唯一分解整環，因為6可以以兩種方式在 <math>\mathbb{Z}[\sqrt{-5}]</math> 中表成整數乘積：<math>2\times 3</math> 和 <math>(1+\sqrt{-5})(1-\sqrt{-5})</math>。
  2. ^ 此指埃爾米特多項式的費馬偽質數
  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<math>\left\langle\sqrt{R},Z_R\right\rangle</math>中，<math>R</math>為負二、<math>Z_R</math>為二的情況<math>\left\langle\pm \mathrm i \sqrt{2},Z_2\right\rangle</math>[62]
  5. ^ 平方根的主值即<math>\sqrt{-x} = \pm i \sqrt{x}</math>取正的值，對於負二而言，即<math>\sqrt{-2} = i \sqrt{2}</math>[註 4][62][64][65][66]

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