−3

-3
	数表 — 整数 << −10 −9‍ −8‍ −7 −6 −5‍ −4 −3 −2 −1 >>
命名
小写	负三
大写	负参
序数词	第负三; negative third
识别
种类	整数
性质
质因数分解	一般不做质因数分解
因数	1、3
绝对值	3
相反数	3
表示方式
值	-3
算筹
二进制	−11(2)
三进制	−10(3)
四进制	−3(4)
五进制	−3(5)
八进制	−3(8)
十二进制	−3(12)
十六进制	−3(16)
	查; 论; 编;

在数学中，负三记作 $-3$ ，是介于负四与负二之间的整数，为 $3$ 的加法逆元或相反数^[1]^:22^[2]，即其与三的和为零^[3]，偶尔会被视为3的逆反词或相对概念^[4]。日常生活中通常不会用负三来计量事物，例如无法具体地描述何谓负三头牛^[4]或持有负三颗苹果^[5]。

Quick Facts ← −4 −3 −2 →, 命名 ...

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负三经常在讯号处理领域被提及，因为负三分贝约为能量的一半^[6]。因此，负三分贝又称为半能点^[7]，经常在滤波器、滤光器和放大器^[8]中使用^[9]。在国际单位制基本单位的表示法中，负三偶尔也会做为幂次来表达立方倒数，比如密度的单位kg･m^-3^[10]。

性质

负三为第二大的负奇数。最大的负奇数为负一，而负三为负一的三倍^[11]。
负三与无理数 $10\log _{10}\left({\tfrac {1}{2}}\right)\approx -3.0103$ 的值十分接近^[12]，因此在讯号处理领域中经常使用负三分贝代表能量为一半的情况^[6]。
负三是最大的负基本判别式（英语：Fundamental discriminant）^[13]，同时，在2-rank为0时，负三是绝对值最小的基本判别式^[14]。
负三能使连续三个奇数的乘积加一为平方数。有这种性质的奇数只有-3和1，而所有满足n(n+2)(n+4)+1为平方数的整数只有11个，分别为-4, -3, -2, 0, 1, 2, 8, 10, 18, 112, 1272^[15]。
负三能使二次域 $\mathbb {Q} [{\sqrt {d}}]$ $\mathbb {Q} [{\sqrt {d}}]$ 的类数为1，即 $\mathbb {Q} [{\sqrt {-3}}]$ $\mathbb {Q} [{\sqrt {-3}}]$ 的类数为1，亦即其整数环为唯一分解整环^{[注 1]}^[16]，且这个二次域在复平面上形成了一个六角网格，每个六边形又可分成6个三角形（三角网格）^[17]^:289。
- 而根据史塔克-黑格纳理论（英语：Stark–Heegner theorem），包含负三，有此性质的负数只有9个^[18]^[17]^:295^[19]^[20]，其对应的自然数称为黑格纳数^[21]。
- 此外负三也能使二次域 $\mathbb {Q} [{\sqrt {d}}]$ 成为简单欧几里得整环（simply Euclidean fields，或称欧几里得范数整环，Norm-Euclidean fields）^[22]，即 $\mathbb {Q} [{\sqrt {-3}}]$ 为简单欧几里得整环。有此性质的负数只有-11, -7, -3, -2, -1（OEIS数列A048981）^[23]。若放宽条件，则-15也能列入^[24]^[25]。
- 若考虑正数，则-3是第七个有此性质的数，前一个是-7、下一个是-2^[16]^[26]。
负三与负三的乘积为正九^[27]，即负三的平方为九^[28]，因此负三为九的平方根之一，即九的负平方根。^{[注 2]}
现有两数i和j，i和j的乘积与六倍i和j的和相等，且其和与i、j皆为整数的结果只有8个解，负三是其中之一^[31]。
负三为四维超立方体（或四维超方形）下闭集合中欧拉示性数的最小值^[32]。

负三的因数

负三的因数有-3, -1, 1和3^[33]，这些因数与3的因数相同。在质因数分解中，虽然能够透过将负一提出来完成质因数分解^[34]^[35]，即 $-3=\,$ $-1\times 3$ ，然而算术基本定理一般以探讨正整数的质因数分解为主^[16]，因此一般不会对负的整数进行质因数分解。^[36]

负三次幂

若一数的幂为负三次，则其可以视为立方的倒数，例如日常生活中常用的密度 CGS制单位g/cm³^[37]，其因此可以表示为质量乘以长度的立方倒数，计为ML^-3，此时负三用以表示立方的倒数^[38]。

而立方倒数中的相关议题还有立方倒数和。自然数的负三次次方和（立方倒数和）会收敛并趋近于阿培里常数，即：

$\sum _{n=1}^{\infty }{n^{-3}}$ =　 ${\frac {\pi ^{2}}{7}}\left\{1-4\sum _{k=1}^{\infty }{\frac {\zeta (2k)}{(2k+1)(2k+2)2^{2k}}}\right\}\approx 1.202056903$ ^[39]

即全体自然数的负三次方和会收敛在这个数。其值约为1.202056903。同时其也是Zeta函数代入3的结果^[39]。

表示方法

负三通常以在3前方加入负号表示^[1]^:28^[40]，通常称为“负三”或大写“负叁”、“负叁”或“负参”，而在某些场合中，会以“零下三”表达-3，例如在表达温度时^[41]^[42]。而在英语中通常以negative three（负三）表示，比较不会以minus three（减三）表示^[43]。

在二进制时，尤其是计算机运算，负数的表示通常会以二补数来表示^[44]，即将所有位数填上1，再向下减。此时，负三计为“......11111101₍₂₎”，例如，在八位元的二补数二进制中，负三会以“11111101₍₂₎”表示，正三会以“00000011₍₂₎”；而在使用负号的表示法中，负三计为“-11₍₂₎”，亦有在最高位填1表示其为负之表示法，此时负三表示为“10000011₍₂₎”^[45]。

在其他领域中

当分贝数为负三时，能量约为一半，又称为半能点^[7]。
智能不足轻度与中度的分界为智力测验平均值的负三个标准差上^[46]
关于十的负三次幂10^-3，其为SI前缀之一，可以用m （Milli）表示。^[47]例如：1毫米为10^-3米、1毫克为10^-3克^[48]。
- 同时10^-3也能代表毫^[49]，以及日本的单位毛（日语：毛 (数)）^[50]。
密度的因次是ML^-3，对应的SI制单位可以表示为kg･m^-3。^[10]^[51] 而加加速度的因次与单位也能用负三次幂表示，其因次计为LT^-3、对应的单位可以用m･s^-3表示。^[52]
部分纪年方法或计算机程序^{[注 3]}容许负值的公元年，此时负三年代表的意义为公元前4年^[54]，同理，对世纪而言负三世纪代表公元前4世纪。^[55]
《-3℃》为岩井由纪子1987年发行的单曲。^[56]
火星^[57]和木星^[58]有时会被归类在负三等星。此外负三等星亦用于火流星的定义：比负三等星亮的流星称为火流星^[59]。
- 当金星位于相对于地球上的太阳背光位置时，其平均视星等约为负三等。^[57]而金星实际上的视星等会在−4.92等和−2.98等之间变动，平均约在−4.14等左右。^[60]
协调世界时为UTC−3表示比协调世界时慢3小时。^[61]
硫酸两个pKa，分别是−3.0和1.99。^[62]^[63]
3-氟丙烯的沸点是−3 °C。^[64]

参见

3
-3 dB（半能点）
前3年

注释

[注 1]
当d<0时，若 $\mathbb {Q} [{\sqrt {d}}]$ 的整数环为唯一分解整环，就表示 $\mathbb {Q} [{\sqrt {d}}]$ 的数字都只有一种因数分解方式，例如 $\mathbb {Q} [{\sqrt {-5}}]$ 的整数环不是唯一分解整环，因为6可以以两种方式在 $\mathbb {Z} [{\sqrt {-5}}]$ 中表成整数乘积： $2\times 3$ 和 $(1+{\sqrt {-5}})(1-{\sqrt {-5}})$ 。
[注 2]
三的平方为九、负三的平方亦为九，故两者皆为九的平方根^[29]^[30]
[注 3]
许多计算机程式库会实作零年的功能，例如Perl CPAN 的 DateTime module^[53]。

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[16] [注 1]
当d<0时，若 $\mathbb {Q} [{\sqrt {d}}]$ 的整数环为唯一分解整环，就表示 $\mathbb {Q} [{\sqrt {d}}]$ 的数字都只有一种因数分解方式，例如 $\mathbb {Q} [{\sqrt {-5}}]$ 的整数环不是唯一分解整环，因为6可以以两种方式在 $\mathbb {Z} [{\sqrt {-5}}]$ 中表成整数乘积： $2\times 3$ 和 $(1+{\sqrt {-5}})(1-{\sqrt {-5}})$ 。

[32] [注 2]
三的平方为九、负三的平方亦为九，故两者皆为九的平方根^[29]^[30]

[56] [注 3]
许多计算机程式库会实作零年的功能，例如Perl CPAN 的 DateTime module^[53]。

[Kilhamn,_Cecilia_2011-1] [1]
Kilhamn, Cecilia. Making Sense of Negative Numbers (Ph.D.论文). University of Gothenburg. 2011. doi:10.13140/RG.2.1.1575.0649.

[2] [2]
Glaeser Georges. Épistémologie des nombres relatifs. Recherches en Didáctique des mathématique: pp.303–346.

[3] [3]
Negative Numbers: Overview § The number -3 (negative three). eduplace.com. [2020-03-20]. （原始内容存档于2012-04-04）.

[book_2015你沒聽過的邏輯課-4] [4]
你沒聽過的邏輯課: 探索魔術、博奕、運動賽事背後的法則. Learn系列. 时报. 2015: 203. ISBN 9789571363189.

[book_martinez2006negative-5] [5]
Martinez, A.A. Negative Math: How Mathematical Rules Can be Positively Bent. Princeton University Press. 2006: 14. ISBN 9780691123097. LCCN 2005043377.

[phys.hawaii.edu_antennas.pdf-6] [6]
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[MATLAB_Power_bandwidth-7] [7]
Power bandwidth - MATLAB powerbw. uk.mathworks.com. [2017-08-05]. （原始内容存档于2021-03-01）.

[8] [8]
Collin Wells. Stability 2 - Op Amps (PDF). TI training Labs. [2020-04-16]. （原始内容存档 (PDF)于2016-02-24）.

[9] [9]
Schlessinger, Monroe. Infrared technology fundamentals 2nd ed., rev. and expanded. New York: M. Dekker. 1995 [2020-03-20]. ISBN 0824792599. （原始内容存档于2017-08-05）.

[SIbrochure8th-10] [10]
International Bureau of Weights and Measures, The International System of Units (SI) (PDF) 8th, 2006, ISBN 92-822-2213-6 （英语）

[11] [11]
Anglin, K.L. CliffsQuickReview Math Word Problems. John Wiley & Sons. 2007: 122. ISBN 9780470197264.

[cox2002fundamentals-12] [12]
Cox, J.F. Fundamentals of Linear Electronics: Integrated and Discrete. Delmar Thomson Learning. 2002: 440. ISBN 9780766830189. LCCN 2001028356.

[13] [13]
Weisstein, Eric W. (编). Fundamental Discriminant. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2020-03-18] （英语）.

[14] [14]
Sloane, N.J.A. (编). Sequence A228251 (Fundamental discriminant of least absolute value with class group of 2-rank n.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[15] [15]
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