−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
素因数分解	一般不做素因数分解
约数	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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负三经常在信号处理领域被提及，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]，因此在信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]。

负三的约数

负三的约数有-3, -1, 1和3^[33]，这些约数与3的约数相同。在素因数分解中，虽然能够透过将负一提出来完成素因数分解^[34]^[35]，即 $-3=\,$ $-1\times 3$ ，然而算术基本定理一般以探讨正整数的素因数分解为主^[16]，因此一般不会对负的整数进行素因数分解。^[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，尤其是电脑运算，负数的表示通常会以补码来表示^[44]，即将所有位数填上1，再向下减。此时，负三计为“......11111101₍₂₎”，例如，在八位元的补码二进制中，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]。
智慧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]
部分纪年方法或电脑程序^{[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 $\mathbb {Q} [{\sqrt {d}}]$ 的数字都只有一种约数分解方式，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]
许多电脑程序库会实现零年的功能，例如Perl CPAN 的 DateTime module^[53]。

参考文献

[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]
Glaeser Georges. Épistémologie des nombres relatifs. Recherches en Didáctique des mathématique: pp.303–346.
[3]
Negative Numbers: Overview § The number -3 (negative three). eduplace.com. [2020-03-20]. （原始内容存档于2012-04-04）.
[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.
[5]
Martinez, A.A. Negative Math: How Mathematical Rules Can be Positively Bent. Princeton University Press. 2006: 14. ISBN 9780691123097. LCCN 2005043377.
[6]
Antenna Introduction / Basics (PDF), [2017-08-08], （原始内容存档 (PDF)于2017-08-28）
[7]
Power bandwidth - MATLAB powerbw. uk.mathworks.com. [2017-08-05]. （原始内容存档于2021-03-01）.
[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]
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）.
[10]
International Bureau of Weights and Measures, The International System of Units (SI) (PDF) 8th, 2006, ISBN 92-822-2213-6 （英语）
[11]
Anglin, K.L. CliffsQuickReview Math Word Problems. John Wiley & Sons. 2007: 122. ISBN 9780470197264.
[12]
Cox, J.F. Fundamentals of Linear Electronics: Integrated and Discrete. Delmar Thomson Learning. 2002: 440. ISBN 9780766830189. LCCN 2001028356.
[13]
Weisstein, Eric W. (编). Fundamental Discriminant. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2020-03-18] （英语）.
[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]
Sloane, N.J.A. (编). Sequence A258692 (Integers n such that n*(n + 2)*(n + 4) + 1 is a perfect square.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
[16]
Hardy, Godfrey Harold; Wright, E. M., An introduction to the theory of numbers Fifth, The Clarendon Press Oxford University Press: 213, 1979 [1938], ISBN 978-0-19-853171-5, MR 0568909
[17]
Riesel, H. Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Birkhäuser Boston, Springer Science & Business Media. 2012. ISBN 9781461202516. LCCN 94027688.
[18]
Conway, John Horton; Guy, Richard K. The Book of Numbers. Springer. 1996: 224. ISBN 0-387-97993-X.
[19]
Stark, H. M., On the gap in the theorem of Heegner (PDF), Journal of Number Theory, 1969, 1: 16–27 [2020-03-20], doi:10.1016/0022-314X(69)90023-7, （原始内容存档 (PDF)于2012-02-11）
[20]
Weisstein, Eric W. (编). Heegner Number. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2020-03-14] （英语）.
[21]
Sloane, N.J.A. (编). Sequence A003173 (Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
[22]
Kyle Bradford and Eugen J. Ionascu, Unit Fractions in Norm-Euclidean Rings of Integers, arXiv, 2014 [2020-03-20], （原始内容存档于2020-03-26）
[23]
LeVeque, William J. Topics in Number Theory, Volumes I and II. New York: Dover Publications. 2002: II:57,81 [1956]. ISBN 978-0-486-42539-9. Zbl 1009.11001.
[24]
Kelly Emmrich and Clark Lyons. Norm-Euclidean Ideals in Galois Cubic Fields (PDF). 2017 West Coast Number Theory Conference. 2017-12-18 [2020-03-20]. （原始内容存档 (PDF)于2020-03-26）.
[25]
Sloane, N.J.A. (编). Sequence A296818 (Squarefree values of n for which the quadratic field Q[ sqrt(n) ] possesses a norm-Euclidean ideal class.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
[26]
Sloane, N.J.A. (编). Sequence A061574 (Simple quadratic fields (i.e., with a unique prime factorization).). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
[27]
Babbage, K.J. Extreme Learning. ScarecrowEducation. 2004: 43. ISBN 9781578861408. LCCN 2004001279.
[28]
GMAT, M. Foundations of GMAT Math. Manhattan Prep GMAT Strategy Guides. Manhattan Prep Publishing. 2011: 73. ISBN 9780979017599.
[29]
Square Root Calculator. calculatorsoup.com. [2020-04-25]. （原始内容存档于2017-05-24）. For example, the square roots of 9 are -3 and +3, since (-3)² = (+3)² = 9.
[30]
Squares and Square Roots, § Negatives. mathsisfun.com. [2020-04-25]. （原始内容存档于2020-08-12）. square root of 9 could be −3 or +3
[31]
Sloane, N.J.A. (编). Sequence A307179 (Numbers k such that k = i*j = 6*i + j, where i and j are integers). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
[32]
Sloane, N.J.A. (编). Sequence A214283 (Smallest Euler characteristic of a downset on an n-dimensional cube). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
[33]
Factors of a Negative Number. sciencing.com. 2018-03-18 [2020-03-20]. （原始内容存档于2017-07-01）.
[34]
José Luis Gómez Pardo. Introduction to Cryptography with Maple. SpringerLink : Bücher. Springer Berlin Heidelberg. 2012: 336. ISBN 9783642321665. LCCN 2012944964.
[35]
Bard, G.V. Sage for Undergraduates. American Mathematical Society. 2015: 269. ISBN 9781470411114. LCCN 14033572.
[36]
Nathanson, M. B. Elementary Methods in Number Theory. Graduate Texts in Mathematics 195. Springer-Verlag. 2000 [2022-08-24]. ISBN 0-387-98912-9. MR 1732941. Zbl 0953.11002. （原始内容存档于2020-09-23）.
[37]
What is a gram per cubic centimeter [g/cm³], a unit of density. AVCalc. [2019-05-18]. （原始内容存档于2019-05-18）（英语）.
[38]
Gram Per Cubic Centimeter. eFunda. [2019-05-18]. （原始内容存档于2019-05-18）（英语）.
[39]
Wedeniwski, Sebastian, Simon Plouffe , 编, The Value of Zeta(3) to 1,000,000 places, Project Gutenberg, 2001 [2020-03-20], （原始内容存档于2021-10-23）
[40]
James W. Brennan. The Real Number System. jamesbrennan.org. [2020-04-16]. （原始内容存档于2020-02-19）.
[41]
別以為印度沒有冬天！零下三度，他們靠這些取暖. 天下杂志. 2019年11月17日.
[42]
Dummies, C. Years 6 - 8 Maths For Students. Wiley. 2015: 25. ISBN 9780730326809.
[43]
Alshwaikh, Jehad and Adler, Jill. Researchers and teachers as learners in Lesson Study. 2017-04: 10. ISBN 978-0-9922269-4-7.
[44]
E.g. Section 4.2.1 in Intel 64 and IA-32 Architectures Software Developer's Manual, Signed integers are two's complement binary values that can be used to represent both positive and negative integer values., Volume 1: Basic Architecture, 2006-11
[45]
Ryan Chadwick. Binary Negative Numbers. ryanstutorials.net. [2020-04-16]. （原始内容存档于2017-08-17）.
[46]
兒童發展與輔導. 教师检验.教师甄试丛书. 志光教育文化. 2005: 1-10,1-11. ISBN 9789861280219.
[47]
AdvancedPhysics Summer Assignment. (PDF). ponderisd.net.
[48]
單位符號用語. tlri.gov.tw. [2020-04-25]. （原始内容存档于2017-08-01）.
[49]
鲁特. 比特幣精粹. Bai xiang wen hua shi ye you xian gong si. 2018. ISBN 9789863587163.
[50]
数の単位. ashiya.ne.jp. [2020-04-16]. （原始内容存档于2017-08-27）.
[51]
The National Aeronautic and Atmospheric Administration's Glenn Research Center. Gas Density Glenn research Center. grc.nasa.gov. [2013-04-09]. （原始内容存档于2013-04-14）.
[52]
Jazar, R.N. Approximation Methods in Science and Engineering. : 21. ISBN 9781071604809.
[53]
Floating DateTimes. [2020-03-26]. （原始内容存档于2020-03-05）.
[54]
Espenak, Fred. Year Dating Conventions. NASA Eclipse Web Site. NASA. [2009-02-19]. （原始内容存档于2009-02-08）.
[55]
Richards, E. G. Calendars. Urban, Sean E.; Seidelmann, P. Kenneth (编). Explanatory Supplement to the Astronomical Almanac 3. Mill Valley, CA: Univ Science Books. 2013: 591. ISBN 1-891389-85-8.
[56]
あの曲も!? 作詞家・及川眠子の集大成「ネコイズム」2・21発売「淋しい熱帯魚」「残酷な天使のテーゼ」ほかヒット曲ずらり. tvlife.jp. 2017-12-31 [2020-03-20]. （原始内容存档于2020-09-22）.
[57]
Mallama, A. Planetary magnitudes. Sky & Telescope. 2011, 121 (1): 51–56.
[58]
Silverman, S.M. and Mullen, E.G. and Air Force Cambridge Research Laboratories (U.S.). Sky Brightness During Eclipses: A Compendium from the Literature. AFCRL TR. Air Force Cambridge Research Laboratories, Air Force Systems Command, United States Air Force. 1974: 27-29.
[59]
火流星概說. vm.nthu.edu.tw. [2020-04-25]. （原始内容存档于2019-12-24）.
[60]
Mallama, Anthony; Hilton, James L. Computing apparent planetary magnitudes for The Astronomical Almanac. Astronomy and Computing. 2018-10, 25: 10–24. Bibcode:2018A&C....25...10M. arXiv:1808.01973 . doi:10.1016/j.ascom.2018.08.002.
[61]
Newfoundland-Labrador Time Zone – Newfoundland-Labrador Current Local Time – Daylight Saving Time. TimeTemperature.com. [2012-10-26]. （原始内容存档于2012-10-23）.
[62]
Wenyi Zhao. Handbook for Chemical Process Research and Development. CRC Press, 2016. 2.1.1.2 Sulfuric Acid. ISBN 9781315350202
[63]
Andrew Burrows, John Holman, Andrew Parsons, Gwen Pilling, Gareth Price. Chemistry³: Introducing Inorganic, Organic and Physical Chemistry. OUP Oxford, 2013. pp 329. The strengths of oxoacids. ISBN 9780199691852
[64]
Haynes, W. M. (2014). CRC Handbook of Chemistry and Physics 95ed. CRC Press. ISBN 97814822-08689.

[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 $\mathbb {Q} [{\sqrt {d}}]$ 的数字都只有一种约数分解方式，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]
许多电脑程序库会实现零年的功能，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]
Antenna Introduction / Basics (PDF), [2017-08-08], （原始内容存档 (PDF)于2017-08-28）

[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]
Sloane, N.J.A. (编). Sequence A258692 (Integers n such that n*(n + 2)*(n + 4) + 1 is a perfect square.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[An_introduction_to_the_theory_of_numbers-17] [16]
Hardy, Godfrey Harold; Wright, E. M., An introduction to the theory of numbers Fifth, The Clarendon Press Oxford University Press: 213, 1979 [1938], ISBN 978-0-19-853171-5, MR 0568909

[book_riesel2012prime-18] [17]
Riesel, H. Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. Birkhäuser Boston, Springer Science & Business Media. 2012. ISBN 9781461202516. LCCN 94027688.

[19] [18]
Conway, John Horton; Guy, Richard K. The Book of Numbers. Springer. 1996: 224. ISBN 0-387-97993-X.

[20] [19]
Stark, H. M., On the gap in the theorem of Heegner (PDF), Journal of Number Theory, 1969, 1: 16–27 [2020-03-20], doi:10.1016/0022-314X(69)90023-7, （原始内容存档 (PDF)于2012-02-11）

[21] [20]
Weisstein, Eric W. (编). Heegner Number. at MathWorld--A Wolfram Web Resource. Wolfram Research, Inc. [2020-03-14] （英语）.

[22] [21]
Sloane, N.J.A. (编). Sequence A003173 (Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[misc_bradford2014unit-23] [22]
Kyle Bradford and Eugen J. Ionascu, Unit Fractions in Norm-Euclidean Rings of Integers, arXiv, 2014 [2020-03-20], （原始内容存档于2020-03-26）

[24] [23]
LeVeque, William J. Topics in Number Theory, Volumes I and II. New York: Dover Publications. 2002: II:57,81 [1956]. ISBN 978-0-486-42539-9. Zbl 1009.11001.

[25] [24]
Kelly Emmrich and Clark Lyons. Norm-Euclidean Ideals in Galois Cubic Fields (PDF). 2017 West Coast Number Theory Conference. 2017-12-18 [2020-03-20]. （原始内容存档 (PDF)于2020-03-26）.

[26] [25]
Sloane, N.J.A. (编). Sequence A296818 (Squarefree values of n for which the quadratic field Q[ sqrt(n) ] possesses a norm-Euclidean ideal class.). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[27] [26]
Sloane, N.J.A. (编). Sequence A061574 (Simple quadratic fields (i.e., with a unique prime factorization).). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[28] [27]
Babbage, K.J. Extreme Learning. ScarecrowEducation. 2004: 43. ISBN 9781578861408. LCCN 2004001279.

[gmat2011foundations-29] [28]
GMAT, M. Foundations of GMAT Math. Manhattan Prep GMAT Strategy Guides. Manhattan Prep Publishing. 2011: 73. ISBN 9780979017599.

[30] [29]
Square Root Calculator. calculatorsoup.com. [2020-04-25]. （原始内容存档于2017-05-24）. For example, the square roots of 9 are -3 and +3, since (-3)² = (+3)² = 9.

[31] [30]
Squares and Square Roots, § Negatives. mathsisfun.com. [2020-04-25]. （原始内容存档于2020-08-12）. square root of 9 could be −3 or +3

[33] [31]
Sloane, N.J.A. (编). Sequence A307179 (Numbers k such that k = i*j = 6*i + j, where i and j are integers). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[34] [32]
Sloane, N.J.A. (编). Sequence A214283 (Smallest Euler characteristic of a downset on an n-dimensional cube). The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[35] [33]
Factors of a Negative Number. sciencing.com. 2018-03-18 [2020-03-20]. （原始内容存档于2017-07-01）.

[book_pardo2012introduction-36] [34]
José Luis Gómez Pardo. Introduction to Cryptography with Maple. SpringerLink : Bücher. Springer Berlin Heidelberg. 2012: 336. ISBN 9783642321665. LCCN 2012944964.

[book_bard2015sage-37] [35]
Bard, G.V. Sage for Undergraduates. American Mathematical Society. 2015: 269. ISBN 9781470411114. LCCN 14033572.

[38] [36]
Nathanson, M. B. Elementary Methods in Number Theory. Graduate Texts in Mathematics 195. Springer-Verlag. 2000 [2022-08-24]. ISBN 0-387-98912-9. MR 1732941. Zbl 0953.11002. （原始内容存档于2020-09-23）.

[39] [37]
What is a gram per cubic centimeter [g/cm³], a unit of density. AVCalc. [2019-05-18]. （原始内容存档于2019-05-18）（英语）.

[40] [38]
Gram Per Cubic Centimeter. eFunda. [2019-05-18]. （原始内容存档于2019-05-18）（英语）.

[Zeta(3)-41] [39]
Wedeniwski, Sebastian, Simon Plouffe , 编, The Value of Zeta(3) to 1,000,000 places, Project Gutenberg, 2001 [2020-03-20], （原始内容存档于2021-10-23）

[42] [40]
James W. Brennan. The Real Number System. jamesbrennan.org. [2020-04-16]. （原始内容存档于2020-02-19）.

[43] [41]
別以為印度沒有冬天！零下三度，他們靠這些取暖. 天下杂志. 2019年11月17日.

[dummies2015years-44] [42]
Dummies, C. Years 6 - 8 Maths For Students. Wiley. 2015: 25. ISBN 9780730326809.

[inbook_Alshwaikh_inbook-45] [43]
Alshwaikh, Jehad and Adler, Jill. Researchers and teachers as learners in Lesson Study. 2017-04: 10. ISBN 978-0-9922269-4-7.

[46] [44]
E.g. Section 4.2.1 in Intel 64 and IA-32 Architectures Software Developer's Manual, Signed integers are two's complement binary values that can be used to represent both positive and negative integer values., Volume 1: Basic Architecture, 2006-11

[47] [45]
Ryan Chadwick. Binary Negative Numbers. ryanstutorials.net. [2020-04-16]. （原始内容存档于2017-08-17）.

[book_2005兒童發展與輔導-48] [46]
兒童發展與輔導. 教师检验.教师甄试丛书. 志光教育文化. 2005: 1-10,1-11. ISBN 9789861280219.

[49] [47]
AdvancedPhysics Summer Assignment. (PDF). ponderisd.net.

[50] [48]
單位符號用語. tlri.gov.tw. [2020-04-25]. （原始内容存档于2017-08-01）.

[2018比特幣精粹-51] [49]
鲁特. 比特幣精粹. Bai xiang wen hua shi ye you xian gong si. 2018. ISBN 9789863587163.

[52] [50]
数の単位. ashiya.ne.jp. [2020-04-16]. （原始内容存档于2017-08-27）.

[53] [51]
The National Aeronautic and Atmospheric Administration's Glenn Research Center. Gas Density Glenn research Center. grc.nasa.gov. [2013-04-09]. （原始内容存档于2013-04-14）.

[jazarapproximation-54] [52]
Jazar, R.N. Approximation Methods in Science and Engineering. : 21. ISBN 9781071604809.

[55] [53]
Floating DateTimes. [2020-03-26]. （原始内容存档于2020-03-05）.

[nasa-eclipses-57] [54]
Espenak, Fred. Year Dating Conventions. NASA Eclipse Web Site. NASA. [2009-02-19]. （原始内容存档于2009-02-08）.

[58] [55]
Richards, E. G. Calendars. Urban, Sean E.; Seidelmann, P. Kenneth (编). Explanatory Supplement to the Astronomical Almanac 3. Mill Valley, CA: Univ Science Books. 2013: 591. ISBN 1-891389-85-8.

[59] [56]
あの曲も!? 作詞家・及川眠子の集大成「ネコイズム」2・21発売「淋しい熱帯魚」「残酷な天使のテーゼ」ほかヒット曲ずらり. tvlife.jp. 2017-12-31 [2020-03-20]. （原始内容存档于2020-09-22）.

[MallamaSky-60] [57]
Mallama, A. Planetary magnitudes. Sky & Telescope. 2011, 121 (1): 51–56.

[book_silverman1974sky-61] [58]
Silverman, S.M. and Mullen, E.G. and Air Force Cambridge Research Laboratories (U.S.). Sky Brightness During Eclipses: A Compendium from the Literature. AFCRL TR. Air Force Cambridge Research Laboratories, Air Force Systems Command, United States Air Force. 1974: 27-29.

[62] [59]
火流星概說. vm.nthu.edu.tw. [2020-04-25]. （原始内容存档于2019-12-24）.

[Mallama_and_Hilton-63] [60]
Mallama, Anthony; Hilton, James L. Computing apparent planetary magnitudes for The Astronomical Almanac. Astronomy and Computing. 2018-10, 25: 10–24. Bibcode:2018A&C....25...10M. arXiv:1808.01973 . doi:10.1016/j.ascom.2018.08.002.

[ttnl-64] [61]
Newfoundland-Labrador Time Zone – Newfoundland-Labrador Current Local Time – Daylight Saving Time. TimeTemperature.com. [2012-10-26]. （原始内容存档于2012-10-23）.

[65] [62]
Wenyi Zhao. Handbook for Chemical Process Research and Development. CRC Press, 2016. 2.1.1.2 Sulfuric Acid. ISBN 9781315350202

[66] [63]
Andrew Burrows, John Holman, Andrew Parsons, Gwen Pilling, Gareth Price. Chemistry³: Introducing Inorganic, Organic and Physical Chemistry. OUP Oxford, 2013. pp 329. The strengths of oxoacids. ISBN 9780199691852

[67] [64]
Haynes, W. M. (2014). CRC Handbook of Chemistry and Physics 95ed. CRC Press. ISBN 97814822-08689.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[注 1]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[注 2]

[31]

[32]

[33]

[34]

[35]

[36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

[注 3]

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

[62]

[63]

[64]

[29]

[30]

[53]