−2 - 维基百科,自由的百科全书
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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 | ||||
值 | -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 | ||||||
↓ |
在數學中,負二是距離原點兩個單位的負整數[1],记作−2[2]或−2[3],是2的加法逆元或相反數,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。
- 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]。
- 負二能使二次域的類数為1,亦即其整數環為唯一分解整環[註 1][19]。而根據史塔克-黑格纳理論,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]。
- 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[31],即[32]。
- [34],同時滿足,即。此外,當為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中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,其中為高斯質數[40]、為虛數單位。
負二的冪
[编辑]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]。
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]:
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[53]。
負二次冪
[编辑]![]() | 由于已知的技术原因,图表暂时不可用。带来不便,我们深表歉意。 |
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]:
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][62][64][65][66]。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[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
[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[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
[编辑]註釋
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