−3:修订间差异
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{{整數|質因數分解=一般不做質因數分解 |
{{整數 |
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|num = -3 |
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|質因數分解=一般不做質因數分解 |
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|因數=1、3 |
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⚫ | 在[[數學]]中,'''負三'''記作{{Math|'''−3'''}},是介於負四與[[-2|負二]]之間的整數,為[[3|{{Math|3}}]]的[[加法逆元]]或[[相反數]]<ref name="Kilhamn, Cecilia 2011">{{Cite thesis |degree=Ph.D. |title=Making Sense of Negative Numbers|author=Kilhamn, Cecilia |year=2011 |publisher=University of Gothenburg |accessdate= | doi=10.13140/RG.2.1.1575.0649 }}</ref>{{rp|22}}<ref>{{Cite journal | author=Glaeser Georges| journal = Recherches en Didáctique des mathématique | title=Épistémologie des nombres relatifs | volume=2 | issue=3|pages=pp.303-346}}</ref>,即其與三的和為零<ref>{{Cite web| title = Negative Numbers: Overview § The number -3 (negative three)| url = https://www.eduplace.com/math/mw/background/5/05/te_5_05_overview.html| publisher = eduplace.com| access-date = 2020-03-20| archive-url = https://web.archive.org/web/20120404014124/http://www.eduplace.com/math/mw/background/5/05/te_5_05_overview.html| archive-date = 2012-04-04| dead-url = |
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|絕對值=[[3]] |
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|相反數=[[3]] |
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}} |
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⚫ | 在[[數學]]中,'''負三'''記作{{Math|'''−3'''}},是介於負四與[[-2|負二]]之間的[[整數]],為[[3|{{Math|3}}]]的[[加法逆元]]或[[相反數]]<ref name="Kilhamn, Cecilia 2011">{{Cite thesis |degree=Ph.D. |title=Making Sense of Negative Numbers|author=Kilhamn, Cecilia |year=2011 |publisher=University of Gothenburg |accessdate= | doi=10.13140/RG.2.1.1575.0649 }}</ref>{{rp|22}}<ref>{{Cite journal | author=Glaeser Georges| journal = Recherches en Didáctique des mathématique | title=Épistémologie des nombres relatifs | volume=2 | issue=3|pages=pp.303-346}}</ref>,即其與三的和為零<ref>{{Cite web| title = Negative Numbers: Overview § The number -3 (negative three)| url = https://www.eduplace.com/math/mw/background/5/05/te_5_05_overview.html| publisher = eduplace.com| access-date = 2020-03-20| archive-url = https://web.archive.org/web/20120404014124/http://www.eduplace.com/math/mw/background/5/05/te_5_05_overview.html| archive-date = 2012-04-04| dead-url = yes}}</ref>,偶爾會被視為3的[[逆反詞]]或相對概念<ref name="book 2015你沒聽過的邏輯課">{{Cite book |
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|title=你沒聽過的邏輯課: 探索魔術、博奕、運動賽事背後的法則 |
|title=你沒聽過的邏輯課: 探索魔術、博奕、運動賽事背後的法則 |
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|isbn=9789571363189 |
|isbn=9789571363189 |
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第12行: | 第18行: | ||
|publisher=時報}}</ref>。日常生活中通常不會用負三來計量事物,例如無法具體地描述何謂負三頭牛<ref name="book 2015你沒聽過的邏輯課"/>或持有負三顆蘋果<ref name="book martinez2006negative">{{Cite book |
|publisher=時報}}</ref>。日常生活中通常不會用負三來計量事物,例如無法具體地描述何謂負三頭牛<ref name="book 2015你沒聽過的邏輯課"/>或持有負三顆蘋果<ref name="book martinez2006negative">{{Cite book |
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|title=Negative Math: How Mathematical Rules Can be Positively Bent |
|title=Negative Math: How Mathematical Rules Can be Positively Bent |
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|url=https://archive.org/details/negativemathhowm0000mart_l0a8 |
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|author=Martinez, A.A. |
|author=Martinez, A.A. |
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|isbn=9780691123097 |
|isbn=9780691123097 |
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|lccn=2005043377 |
|lccn=2005043377 |
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|page=[https://archive.org/details/negativemathhowm0000mart_l0a8/page/n14 14] |
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|page=14 |
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|year=2006 |
|year=2006 |
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|publisher=Princeton University Press}}</ref>。 |
|publisher=Princeton University Press}}</ref>。 |
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負三經常在訊號處理領域被提及,因為負三[[分貝]]約為能量的一半<ref name="phys.hawaii.edu antennas.pdf">{{citation |url=http://www.phys.hawaii.edu/~anita/new/papers/militaryHandbook/antennas.pdf |title=Antenna Introduction / Basics |access-date=2017-08-08 |archive-url=https://web.archive.org/web/20170828230007/http://www.phys.hawaii.edu/~anita/new/papers/militaryHandbook/antennas.pdf |archive-date=2017-08-28 |dead-url=no }}</ref>。因此,負三分貝又稱為 |
負三經常在[[訊號處理]]領域被提及,因為負三[[分貝]]約為能量的一半<ref name="phys.hawaii.edu antennas.pdf">{{citation |url=http://www.phys.hawaii.edu/~anita/new/papers/militaryHandbook/antennas.pdf |title=Antenna Introduction / Basics |access-date=2017-08-08 |archive-url=https://web.archive.org/web/20170828230007/http://www.phys.hawaii.edu/~anita/new/papers/militaryHandbook/antennas.pdf |archive-date=2017-08-28 |dead-url=no }}</ref>。因此,負三分貝又稱為[[半能點]]<ref name="MATLAB Power bandwidth">{{cite web|title=Power bandwidth - MATLAB powerbw|url=https://uk.mathworks.com/help/signal/ref/powerbw.html?s_tid=gn_loc_drop|website=uk.mathworks.com|accessdate=2017-08-05|archive-date=2021-03-01|archive-url=https://web.archive.org/web/20210301015048/https://uk.mathworks.com/help/signal/ref/powerbw.html?s_tid=gn_loc_drop|dead-url=no}}</ref>,經常在[[濾波器]]、[[滤光器]]和[[放大器]]<ref>{{Cite web | url=https://training.ti.com/system/files/docs/1332%20-%20Stability%202%20-%20slides.pdf | title=Stability 2 - Op Amps | author=Collin Wells | publisher=TI training Labs | accessdate=2020-04-16 | archive-date=2016-02-24 | archive-url=https://web.archive.org/web/20160224090134/https://training.ti.com/system/files/docs/1332%20-%20Stability%202%20-%20slides.pdf | dead-url=no }}</ref>中使用<ref>{{cite book|last1=Schlessinger|first1=Monroe|title=Infrared technology fundamentals|date=1995|publisher=M. Dekker|location=New York|isbn=0824792599|edition=2nd ed., rev. and expanded.|url=https://books.google.co.uk/books?id=QPBQ5w4X8RkC&pg=PA113&lpg=PA113&dq=half-power+point&source=bl&ots=dF_wISrwL2&sig=p3_n5NGCzWo5ws2C61QfrRim7ww&hl=en&sa=X&ved=0ahUKEwjHn5jpl8DVAhWmOsAKHSu9Ad84ChDoAQhYMAg#v=onepage&q=half-power%20point&f=false|access-date=2020-03-20|archive-url=https://web.archive.org/web/20170805182343/https://books.google.co.uk/books?id=QPBQ5w4X8RkC&pg=PA113&lpg=PA113&dq=half-power+point&source=bl&ots=dF_wISrwL2&sig=p3_n5NGCzWo5ws2C61QfrRim7ww&hl=en&sa=X&ved=0ahUKEwjHn5jpl8DVAhWmOsAKHSu9Ad84ChDoAQhYMAg#v=onepage&q=half-power%20point&f=false|archive-date=2017-08-05|dead-url=no}}</ref>。在[[國際單位制]]基本單位的表示法中,負三偶爾也會做為冪次來表達立方倒數,比如[[密度]]的单位kg・m<sup>-3</sup><ref name="SIbrochure8th">{{SIbrochure8th}}</ref>。 |
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== 性質 == |
== 性質 == |
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*負三為第二大的負奇數。最大的負奇數為負一,而負三為負一的三倍<ref>{{Cite book | author=Anglin, K.L. | isbn=9780470197264 | page=122 | publisher=John Wiley & Sons | title=CliffsQuickReview Math Word Problems | year=2007 }}</ref>。 |
*負三為第二大的負奇數。最大的負奇數為[[-1|負一]],而負三為負一的三倍<ref>{{Cite book | author=Anglin, K.L. | isbn=9780470197264 | page=122 | publisher=John Wiley & Sons | title=CliffsQuickReview Math Word Problems | year=2007 }}</ref>。 |
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* 負三與無理數<math>10\log_{10}\left(\tfrac{1}{2}\right) \approx -3.0103</math>的值[[接近整數|十分接近]]<ref name=cox2002fundamentals>{{Cite book |
* 負三與無理數<math>10\log_{10}\left(\tfrac{1}{2}\right) \approx -3.0103</math>的值[[接近整數|十分接近]]<ref name=cox2002fundamentals>{{Cite book |
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|title=Fundamentals of Linear Electronics: Integrated and Discrete |
|title=Fundamentals of Linear Electronics: Integrated and Discrete |
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|url=https://archive.org/details/fundamentalsofli0000coxj |
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|author=Cox, J.F. |
|author=Cox, J.F. |
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|isbn=9780766830189 |
|isbn=9780766830189 |
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|lccn=2001028356 |
|lccn=2001028356 |
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|page=[https://archive.org/details/fundamentalsofli0000coxj/page/440 440] |
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|page=440 |
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|year=2002 |
|year=2002 |
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|publisher=Delmar Thomson Learning}}</ref>,因此在訊號處理領域中經常使用負三[[分貝]]代表能量為一半的情況<ref name="phys.hawaii.edu antennas.pdf"/>。 |
|publisher=Delmar Thomson Learning}}</ref>,因此在訊號處理領域中經常使用負三[[分貝]]代表能量為一半的情況<ref name="phys.hawaii.edu antennas.pdf"/>。 |
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* 負三是最大的負{{link-en|基本判别式|Fundamental discriminant}}<ref>{{Cite MathWorld | title=Fundamental Discriminant | urlname=FundamentalDiscriminant | accessdate=2020-03-18 }}</ref>,同時,在2-rank為0時,負三是絕對值最小的基本判别式<ref>{{Cite OEIS|sequencenumber=A228251|name=Fundamental discriminant of least absolute value with class group of 2-rank n.}}</ref>。 |
* 負三是最大的負{{link-en|基本判别式|Fundamental discriminant}}<ref>{{Cite MathWorld | title=Fundamental Discriminant | urlname=FundamentalDiscriminant | accessdate=2020-03-18 }}</ref>,同時,在2-rank為0時,負三是絕對值最小的基本判别式<ref>{{Cite OEIS|sequencenumber=A228251|name=Fundamental discriminant of least absolute value with class group of 2-rank n.}}</ref>。 |
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* 負三能使連續三個奇數的乘積加一為平方數。有這種性質的奇數只有[[-3]]和[[1]],而所有滿足n(n+2)(n+4)+1為平方數的整數只有11個,分別為-4, -3, -2, 0, 1, 2, 8, 10, 18, 112, 1272<ref>{{Cite OEIS|sequencenumber=A258692|name=Integers n such that n*(n + 2)*(n + 4) + 1 is a perfect square.}}</ref>。 |
* 負三能使連續三個奇數的乘積加一為平方數。有這種性質的奇數只有[[-3]]和[[1]],而所有滿足n(n+2)(n+4)+1為平方數的整數只有11個,分別為-4, -3, -2, 0, 1, 2, 8, 10, 18, 112, 1272<ref>{{Cite OEIS|sequencenumber=A258692|name=Integers n such that n*(n + 2)*(n + 4) + 1 is a perfect square.}}</ref>。 |
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* 負三能使[[二次域]]<math>\mathbb{Q}[\sqrt{d}]</math>的[[理想類群|類数]]為1,即<math>\mathbb{Q}[\sqrt{-3}]</math>的[[理想類群|類数]]為1,亦即其[[整數環]]為[[唯一分解整環]]<ref group="註">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>。</ref><ref name="An introduction to the theory of numbers">{{Citation | last1=Hardy | first1=Godfrey Harold | author1-link=G. H. Hardy | last2=Wright | first2=E. M. | title=An introduction to the theory of numbers | origyear=1938 | publisher=The Clarendon Press Oxford University Press | edition=Fifth | isbn=978-0-19-853171-5 | mr=568909 | year=1979 | page=213 |
* 負三能使[[二次域]]<math>\mathbb{Q}[\sqrt{d}]</math>的[[理想類群|類数]]為1,即<math>\mathbb{Q}[\sqrt{-3}]</math>的[[理想類群|類数]]為1,亦即其[[整數環]]為[[唯一分解整環]]<ref group="註">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>。</ref><ref name="An introduction to the theory of numbers">{{Citation | last1=Hardy | first1=Godfrey Harold | author1-link=G. H. Hardy | last2=Wright | first2=E. M. | title=An introduction to the theory of numbers | origyear=1938 | publisher=The Clarendon Press Oxford University Press | edition=Fifth | isbn=978-0-19-853171-5 | mr=568909 | year=1979 | page=213 | url=https://archive.org/details/introductiontoth00hard }}</ref>,且這個二次域在[[複平面]]上形成了一個[[正六邊形鑲嵌|六角網格]],每個六邊形又可分成6個[[三角形]]([[正三角形鑲嵌|三角網格]])<ref name="book riesel2012prime"/>{{rp|289}}。 |
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**而根據{{link-en|史塔克-黑格纳理論|Stark–Heegner theorem}},包含負三,有此性 |
**而根據{{link-en|史塔克-黑格纳理論|Stark–Heegner theorem}},包含負三,有此性質的負數只有9個<ref>{{cite book |
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| last = Conway |
| last = Conway |
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| first = John Horton |
| first = John Horton |
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*負三與負三的乘積為正九<ref>{{Cite book | author=Babbage, K.J. | isbn=9781578861408 | lccn=2004001279 | page=43 | publisher=ScarecrowEducation | title=Extreme Learning | year=2004 }}</ref>,即負三的平方為九<ref name="gmat2011foundations">{{Cite book |
*負三與負三的乘積為正九<ref>{{Cite book | author=Babbage, K.J. | isbn=9781578861408 | lccn=2004001279 | page=43 | publisher=ScarecrowEducation | title=Extreme Learning | year=2004 }}</ref>,即負三的平方為九<ref name="gmat2011foundations">{{Cite book |
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|title=Foundations of GMAT Math |
|title=Foundations of GMAT Math |
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|url=https://archive.org/details/manhattangmatstr00gmat_254 |
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|author=GMAT, M. |
|author=GMAT, M. |
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|isbn=9780979017599 |
|isbn=9780979017599 |
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|series=Manhattan Prep GMAT Strategy Guides |
|series=Manhattan Prep GMAT Strategy Guides |
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|page=[https://archive.org/details/manhattangmatstr00gmat_254/page/n280 73] |
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|page=73 |
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|year=2011 |
|year=2011 |
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|publisher=Manhattan Prep Publishing}}</ref>,因此負三為九的平方根之一,即九的負平方根。{{#tag:ref|三的平方為九、負三的平方亦為九,故兩者皆為九的平方根<ref>{{Cite web | url=https://www.calculatorsoup.com/calculators/algebra/squareroots.php | title=Square Root Calculator | publisher=calculatorsoup.com | quote=For example, the square roots of 9 are -3 and +3, since (-3)<sup>2</sup> = (+3)<sup>2</sup> = 9. | accessdate=2020-04-25 | archive-date=2017-05-24 | archive-url=https://web.archive.org/web/20170524012057/http://www.calculatorsoup.com/calculators/algebra/squareroots.php | dead-url=no }}</ref><ref>{{Cite web | url=https://www.mathsisfun.com/square-root.html | title=Squares and Square Roots, § Negatives | publisher=mathsisfun.com | quote=square root of 9 could be −3 or +3 | accessdate=2020-04-25 | archive-date=2020-08-12 | archive-url=https://web.archive.org/web/20200812140828/https://www.mathsisfun.com/square-root.html | dead-url=yes }}</ref>|group="註"}} |
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|publisher=Manhattan Prep Publishing}}</ref>,因此負三為九的平方根之一,即九的負平方根。 |
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*現有兩數i和j,i和j的乘積與六倍i和j的和相等,且其和與i、j皆為整數的結果只有8個解,負三是其中之一<ref>{{Cite OEIS|sequencenumber= |
*現有兩數i和j,i和j的乘積與六倍i和j的和相等,且其和與i、j皆為整數的結果只有8個解,負三是其中之一<ref>{{Cite OEIS|sequencenumber= |
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A307179|name=Numbers k such that k = i*j = 6*i + j, where i and j are integers }}</ref>。 |
A307179|name=Numbers k such that k = i*j = 6*i + j, where i and j are integers }}</ref>。 |
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*負三為[[ |
*負三為[[四維超立方體]](或四維[[超方形]])[[下闭集合]]中[[欧拉示性数]]的最小值<ref>{{Cite OEIS|sequencenumber=A214283|name=Smallest Euler characteristic of a downset on an n-dimensional cube}}</ref>。 |
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=== 負三的因數 === |
=== 負三的因數 === |
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第96行: | 第105行: | ||
|publisher=American Mathematical Society}}</ref>, |
|publisher=American Mathematical Society}}</ref>, |
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即{{質因數分解|number=-3|number class=高斯整數|show number=yes}},然而[[算术基本定理]]一般以探討正整數的[[質因數分解]]為主<ref name="An introduction to the theory of numbers"/>,因此一般不會對負的整數進行質因數分解。<ref>{{cite book |
即{{質因數分解|number=-3|number class=高斯整數|show number=yes}},然而[[算术基本定理]]一般以探討正整數的[[質因數分解]]為主<ref name="An introduction to the theory of numbers"/>,因此一般不會對負的整數進行質因數分解。<ref>{{cite book |
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|last1 = Nathanson |
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|first1 = M. B. |
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|year = 2000 |
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|title = Elementary Methods in Number Theory |
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|series = Graduate Texts in Mathematics |
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|volume = 195 |
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|url = https://books.google.com/books?id=sE7lBwAAQBAJ |
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|publisher = Springer-Verlag |
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|isbn = 0-387-98912-9 |
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|mr = 1732941 |
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|zbl = 0953.11002 |
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|ref = harv |
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|access-date = 2022-08-24 |
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|archive-date = 2020-09-23 |
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|archive-url = https://web.archive.org/web/20200923213938/https://books.google.com/books?id=sE7lBwAAQBAJ |
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|dead-url = no |
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}}</ref> |
}}</ref> |
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=== 負三次冪 === |
=== 負三次冪 === |
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{{Anchor|立方的倒數|立方倒數}}若一數的冪為負三次,則其可以視為立方的倒數,例如日常生活中常用的[[密度]][[CGS制]]單位g/cm<sup>3</sup><ref>{{Cite web|url=https://www.aqua-calc.com/what-is/density/gram-per-cubic-centimeter|title=What is a gram per cubic centimeter [g/cm³], a unit of density|publisher=AVCalc|language=en|accessdate=2019-05-18|archive-url=https://web.archive.org/web/20190518104831/https://www.aqua-calc.com/what-is/density/gram-per-cubic-centimeter|archive-date=2019-05-18|dead-url=no}}</ref>,其因此可以表示為質量乘以長度的立方倒數,計為{{nowrap|ML<sup>-3</sup>}},此時負三用以表示立方的倒數<ref>{{Cite web|url=http://www.efunda.com/glossary/units/units--density--gram_per_cubic_centimeter.cfm|title=Gram Per Cubic Centimeter|publisher=eFunda|language=en|accessdate=2019-05-18|archive-url=https://web.archive.org/web/20190518154659/http://www.efunda.com/glossary/units/units--density--gram_per_cubic_centimeter.cfm|archive-date=2019-05-18|dead-url=no}}</ref> |
{{Anchor|立方的倒數|立方倒數}}若一數的冪為負三次,則其可以視為立方的倒數,例如日常生活中常用的[[密度]][[CGS制]]單位g/cm<sup>3</sup><ref>{{Cite web|url=https://www.aqua-calc.com/what-is/density/gram-per-cubic-centimeter|title=What is a gram per cubic centimeter [g/cm³], a unit of density|publisher=AVCalc|language=en|accessdate=2019-05-18|archive-url=https://web.archive.org/web/20190518104831/https://www.aqua-calc.com/what-is/density/gram-per-cubic-centimeter|archive-date=2019-05-18|dead-url=no}}</ref>,其因此可以表示為質-{}-量乘以長度的立方倒數,計為{{nowrap|ML<sup>-3</sup>}},此時負三用以表示立方的倒數<ref>{{Cite web|url=http://www.efunda.com/glossary/units/units--density--gram_per_cubic_centimeter.cfm|title=Gram Per Cubic Centimeter|publisher=eFunda|language=en|accessdate=2019-05-18|archive-url=https://web.archive.org/web/20190518154659/http://www.efunda.com/glossary/units/units--density--gram_per_cubic_centimeter.cfm|archive-date=2019-05-18|dead-url=no}}</ref>。 |
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而立方倒數中的相關議題還有立方倒數和。自然數的負三次次方和(立方倒數和)會收斂並趨近於[[阿培里常数]],即: |
而立方倒數中的相關議題還有立方倒數和。自然數的負三次次方和(立方倒數和)會收斂並趨近於[[阿培里常数]],即: |
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| publisher = Project Gutenberg |
| publisher = Project Gutenberg |
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| url = http://www.gutenberg.org/cache/epub/2583/pg2583.html |
| url = http://www.gutenberg.org/cache/epub/2583/pg2583.html |
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| accessdate = 2020-03-20 |
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| archive-date = 2021-10-23 |
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| archive-url = https://web.archive.org/web/20211023215420/https://www.gutenberg.org/cache/epub/2583/pg2583.html |
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即全體自然數的負三次方[[求和符号|和]]會收斂在這個數。其值約為1.202056903。同時其也是[[黎曼ζ函數|Zeta函數]]代入3的結果<ref name="Zeta(3)"/>。 |
即全體自然數的負三次方[[求和符号|和]]會收斂在這個數。其值約為1.202056903。同時其也是[[黎曼ζ函數|Zeta函數]]代入3的結果<ref name="Zeta(3)"/>。 |
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== 表示方法 == |
== 表示方法 == |
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負三通常以在3前方加入負號表示<ref name="Kilhamn, Cecilia 2011"/>{{rp|28}}<ref>{{Cite web | url = https://www.jamesbrennan.org/algebra/numbers/real_number_system.htm | website=jamesbrennan.org | title= |
負三通常以在3前方加入負號表示<ref name="Kilhamn, Cecilia 2011"/>{{rp|28}}<ref>{{Cite web | url = https://www.jamesbrennan.org/algebra/numbers/real_number_system.htm | website = jamesbrennan.org | title = The Real Number System | author = James W. Brennan | accessdate = 2020-04-16 | archive-date = 2020-02-19 | archive-url = https://web.archive.org/web/20200219055112/http://www.jamesbrennan.org/algebra/numbers/real_number_system.htm | dead-url = no }}</ref>,通常稱為「負三」或大寫「負叄」、「負叁」或「負參」,而在某些場合中,會以「零下三」表達-3,例如在表達溫度時<ref>{{Cite Web|title=別以為印度沒有冬天!零下三度,他們靠這些取暖|url=https://www.cw.com.tw/article/article.action?id=5097709|date=2019年11月17日|publisher=[[天下雜誌]]}}</ref><ref name=dummies2015years>{{Cite book |
||
The Real Number System |author=James W. Brennan}}</ref>,通常稱為「負三」或大寫「負叄」、「負叁」或「負參」,而在某些場合中,會以「零下三」表達-3,例如在表達溫度時<ref>{{Cite Web|title=別以為印度沒有冬天!零下三度,他們靠這些取暖|url=https://www.cw.com.tw/article/article.action?id=5097709|date=2019年11月17日|publisher=[[天下雜誌]]}}</ref><ref name=dummies2015years>{{Cite book |
|||
|title=Years 6 - 8 Maths For Students |
|title=Years 6 - 8 Maths For Students |
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|author=Dummies, C. |
|author=Dummies, C. |
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第137行: | 第153行: | ||
|publisher=Wiley}}</ref>。而在英語中通常以negative three(負三)表示,比較不會以minus three(減三)表示<ref name="inbook Alshwaikh inbook">{{Cite book|author=Alshwaikh, Jehad and Adler, Jill|date=2017-04|page=10|title=Researchers and teachers as learners in Lesson Study|isbn=978-0-9922269-4-7}}</ref>。 |
|publisher=Wiley}}</ref>。而在英語中通常以negative three(負三)表示,比較不會以minus three(減三)表示<ref name="inbook Alshwaikh inbook">{{Cite book|author=Alshwaikh, Jehad and Adler, Jill|date=2017-04|page=10|title=Researchers and teachers as learners in Lesson Study|isbn=978-0-9922269-4-7}}</ref>。 |
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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<ref>E.g. {{citation|title=Signed integers are two's complement binary values that can be used to represent both positive and negative integer values.|section=Section 4.2.1 in Intel 64 and IA-32 Architectures Software Developer's Manual|volume=Volume 1: Basic Architecture|date= |
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<ref>E.g. {{citation|title=Signed integers are two's complement binary values that can be used to represent both positive and negative integer values.|section=Section 4.2.1 in Intel 64 and IA-32 Architectures Software Developer's Manual|volume=Volume 1: Basic Architecture|date=2006-11}}</ref>,即將所有位數填上1,再向下減。此時,負三計為「......11111101<sub>(2)</sub>」,例如,在八位元的[[二補數]]二進制中,負三會以「11111101<sub>(2)</sub>」表示,正三會以「00000011<sub>(2)</sub>」;而在使用負號的表示法中,負三計為「-11<sub>(2)</sub>」,亦有在最高位填1表示其為負之表示法,此時負三表示為「10000011<sub>(2)</sub>」<ref>{{Cite web | url=https://ryanstutorials.net/binary-tutorial/binary-negative-numbers.php | title=Binary Negative Numbers | author=Ryan Chadwick | publisher=ryanstutorials.net | accessdate=2020-04-16 | archive-date=2017-08-17 | archive-url=https://web.archive.org/web/20170817160352/http://ryanstutorials.net/binary-tutorial/binary-negative-numbers.php | dead-url=no }}</ref>。 |
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== 在其他領域中 == |
== 在其他領域中 == |
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*當分貝數為負三時,能量約為一半,又稱為 |
*當分貝數為負三時,能量約為一半,又稱為[[半能點]]<ref name="MATLAB Power bandwidth"/>。 |
||
*智能不足輕度與中度的分界為智力測驗平均值的負三個標準差上<ref name="book 2005兒童發展與輔導">{{Cite book |
*智能不足輕度與中度的分界為智力測驗平均值的負三個標準差上<ref name="book 2005兒童發展與輔導">{{Cite book |
||
|title=兒童發展與輔導 |
|title=兒童發展與輔導 |
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第148行: | 第164行: | ||
|year=2005 |
|year=2005 |
||
|publisher=志光教育文化}}</ref> |
|publisher=志光教育文化}}</ref> |
||
* 關於十的負三次冪10<sup>-3 </sup>, 其為[[国际单位制词头|SI前缀]]之一,可以用m ([[毫|Milli]])表示。<ref>{{Cite web|title=AdvancedPhysics Summer Assignment.|url=https://www.ponderisd.net/cms/lib/TX01001056/Centricity/Domain/90/Advanced%20Physics%20Summer%20Assignment.pdf|publisher=ponderisd.net}}</ref>例如:1毫米為10<sup>-3 </sup>米、1毫克為10<sup>-3 </sup>克 |
* 關於十的負三次冪10<sup>-3 </sup>, 其為[[国际单位制词头|SI前缀]]之一,可以用m ([[毫|Milli]])表示。<ref>{{Cite web|title=AdvancedPhysics Summer Assignment.|url=https://www.ponderisd.net/cms/lib/TX01001056/Centricity/Domain/90/Advanced%20Physics%20Summer%20Assignment.pdf|publisher=ponderisd.net}}</ref>例如:1毫米為10<sup>-3 </sup>米、1毫克為10<sup>-3 </sup>克<ref>{{cite web|url=https://www.tlri.gov.tw/Term/Symbole.htm|title=單位符號用語|publisher=tlri.gov.tw|accessdate=2020-04-25|archive-date=2017-08-01|archive-url=https://web.archive.org/web/20170801174359/http://www.tlri.gov.tw/Term/Symbole.htm|dead-url=yes}}</ref>。 |
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** 同時10<sup>-3</sup>也能代表[[中文数字#中文小數單位|毫]]<ref name="2018比特幣精粹">{{Cite book |
** 同時10<sup>-3</sup>也能代表[[中文数字#中文小數單位|毫]]<ref name="2018比特幣精粹">{{Cite book |
||
|title=比特幣精粹 |
|title=比特幣精粹 |
||
第155行: | 第171行: | ||
|year=2018 |
|year=2018 |
||
|author=魯特 |
|author=魯特 |
||
|publisher=Bai xiang wen hua shi ye you xian gong si}}</ref>,以及日本的單位{{link-ja|毛 (數)|毛 (数)|毛}}<ref>{{Cite web | url = http://www.ashiya.ne.jp/nano3.htm | title = 数の単位 | publisher=ashiya.ne.jp}}</ref>。 |
|publisher=Bai xiang wen hua shi ye you xian gong si}}</ref>,以及日本的單位{{link-ja|毛 (數)|毛 (数)|毛}}<ref>{{Cite web | url = http://www.ashiya.ne.jp/nano3.htm | title = 数の単位 | publisher = ashiya.ne.jp | accessdate = 2020-04-16 | archive-date = 2017-08-27 | archive-url = https://web.archive.org/web/20170827082905/http://www.ashiya.ne.jp/nano3.htm | dead-url = no }}</ref>。 |
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*[[密度]]的[[因次]]是ML<sup>-3</sup>,對應的SI制單位可以表示為kg・m<sup>-3</sup>。<ref name="SIbrochure8th"/><ref>{{cite web | url =http://www.grc.nasa.gov/WWW/BGH/fluden.html | title =Gas Density Glenn research Center | author =''[[美国国家航空航天局|The National Aeronautic and Atmospheric Administration's]]'' ''[[Glenn Research Center]]'' | publisher =grc.nasa.gov | archiveurl =https://web.archive.org/web/20130414132531/http://www.grc.nasa.gov/WWW/BGH/fluden.html | archivedate = |
*[[密度]]的[[因次]]是ML<sup>-3</sup>,對應的SI制單位可以表示為kg・m<sup>-3</sup>。<ref name="SIbrochure8th"/><ref>{{cite web | url =http://www.grc.nasa.gov/WWW/BGH/fluden.html | title =Gas Density Glenn research Center | author =''[[美国国家航空航天局|The National Aeronautic and Atmospheric Administration's]]'' ''[[Glenn Research Center]]'' | publisher =grc.nasa.gov | archiveurl =https://web.archive.org/web/20130414132531/http://www.grc.nasa.gov/WWW/BGH/fluden.html | archivedate =2013-04-14 | access-date =2013-04-09 | dead-url =no }}</ref> 而[[加加速度]]的因次與單位也能用負三次冪表示,其因次計為LT<sup>-3</sup>、對應的單位可以用m・s<sup>-3 </sup>表示 。<ref name="jazarapproximation">{{cite book |
||
|title=Approximation Methods in Science and Engineering |
|title=Approximation Methods in Science and Engineering |
||
|author=Jazar, R.N. |
|author=Jazar, R.N. |
||
|isbn=9781071604809 |
|isbn=9781071604809 |
||
|page=21}}</ref> |
|page=21}}</ref> |
||
* 部分紀年方法或計算機程序{{#tag:ref| |
* 部分紀年方法或計算機程序{{#tag:ref|許多計算機程式庫會實作零年的功能,例如Perl CPAN 的 DateTime module<ref>{{Cite web |url=https://metacpan.org/pod/release/DROLSKY/DateTime-1.03/lib/DateTime.pm#Floating-DateTimes |title=Floating DateTimes |access-date=2020-03-26 |archive-url=https://web.archive.org/web/20200305100057/https://metacpan.org/pod/release/DROLSKY/DateTime-1.03/lib/DateTime.pm#Floating-DateTimes |archive-date=2020-03-05 |dead-url=no }}</ref>。|group="註"}}容許負值的[[公元]]年,此時負三年代表的意義為公元[[前4年]]<ref name="nasa-eclipses">{{cite web|url=http://eclipse.gsfc.nasa.gov/SEhelp/dates.html|title=Year Dating Conventions|last=Espenak|first=Fred|work=NASA Eclipse Web Site|publisher=NASA|accessdate=2009-02-19|archiveurl=https://web.archive.org/web/20090208212742/http://eclipse.gsfc.nasa.gov/SEhelp/dates.html|archivedate=2009-02-08|dead-url=no}}</ref>,同理,對世紀而言負三世紀代表[[前4世纪|公元前4世纪]]。<ref>{{Cite book| edition = 3| publisher = Univ Science Books| isbn = 1-891389-85-8| editor1-first = Sean E. | editor1-last= Urban | editor2-first = P. Kenneth | editor2-last= Seidelmann | title = Explanatory Supplement to the Astronomical Almanac| chapter = Calendars | author-first = E. G. | author-last = Richards | page = 591 | location = Mill Valley, CA| date = 2013}}</ref> |
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* 《-3℃》為[[岩井由紀子]]1987年發行的單曲。<ref>{{Cite web|title=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|url=https://www.tvlife.jp/entame/153751|date=2017-12-31|publisher=tvlife.jp}}</ref> |
* 《-3℃》為[[岩井由紀子]]1987年發行的單曲。<ref>{{Cite web|title=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|url=https://www.tvlife.jp/entame/153751|date=2017-12-31|publisher=tvlife.jp|access-date=2020-03-20|archive-date=2020-09-22|archive-url=https://web.archive.org/web/20200922210011/https://www.tvlife.jp/entame/153751|dead-url=no}}</ref> |
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*[[火星]]<ref name="MallamaSky"/>和[[木星]]<ref name="book silverman1974sky">{{Cite book |
*[[火星]]<ref name="MallamaSky"/>和[[木星]]<ref name="book silverman1974sky">{{Cite book |
||
|title=Sky Brightness During Eclipses: A Compendium from the Literature |
|title=Sky Brightness During Eclipses: A Compendium from the Literature |
||
第169行: | 第185行: | ||
|page=27-29 |
|page=27-29 |
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|year=1974 |
|year=1974 |
||
|publisher=Air Force Cambridge Research Laboratories, Air Force Systems Command, United States Air Force}}</ref>有時會被歸類在負三等星。 |
|publisher=Air Force Cambridge Research Laboratories, Air Force Systems Command, United States Air Force}}</ref>有時會被歸類在負三等星。此外負三等星亦用於火流星的定義:比負三等星亮的流星稱為火流星<ref>{{Cite web|url=http://thnf-web.vm.nthu.edu.tw/science/shows/leonids/fireballs.html|title=火流星概說|publisher=vm.nthu.edu.tw|access-date=2020-04-25|archive-url=https://web.archive.org/web/20191224162055/http://thnf-web.vm.nthu.edu.tw/science/shows/leonids/fireballs.html|archive-date=2019-12-24|dead-url=yes}}</ref>。 |
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**當[[金星]]位於相對於地球上的太陽背光位置時,其平均視星等約為負三等。<ref name="MallamaSky">{{cite journal |last=Mallama |first=A. |title=Planetary magnitudes |journal=Sky & Telescope |volume=121 |issue=1 |pages=51–56 |date=2011}} |
**當[[金星]]位於相對於地球上的太陽背光位置時,其平均視星等約為負三等。<ref name="MallamaSky">{{cite journal |last=Mallama |first=A. |title=Planetary magnitudes |journal=Sky & Telescope |volume=121 |issue=1 |pages=51–56 |date=2011}} |
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</ref>而金星實際上的視星等會在−4.92等和−2.98等之間變動,平均約在−4.14等左右。<ref name="Mallama_and_Hilton">{{cite journal |title=Computing apparent planetary magnitudes for The Astronomical Almanac |journal=Astronomy and Computing |first1=Anthony |last1=Mallama |first2=James L. |last2=Hilton |volume=25 |pages=10–24 |date= |
</ref>而金星實際上的視星等會在−4.92等和−2.98等之間變動,平均約在−4.14等左右。<ref name="Mallama_and_Hilton">{{cite journal |title=Computing apparent planetary magnitudes for The Astronomical Almanac |journal=Astronomy and Computing |first1=Anthony |last1=Mallama |first2=James L. |last2=Hilton |volume=25 |pages=10–24 |date=2018-10 |doi=10.1016/j.ascom.2018.08.002 |bibcode=2018A&C....25...10M |arxiv=1808.01973}}</ref> |
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*协调世界时为[[UTC−3]]表示比协调世界时慢3小时。<ref name=ttnl>{{cite web|url=http://www.timetemperature.com/tzca/newfoundland_time_zone.shtml|title=Newfoundland-Labrador Time Zone – Newfoundland-Labrador Current Local Time – Daylight Saving Time|publisher=TimeTemperature.com|accessdate=26 |
*协调世界时为[[UTC−3]]表示比协调世界时慢3小时。<ref name=ttnl>{{cite web|url=http://www.timetemperature.com/tzca/newfoundland_time_zone.shtml|title=Newfoundland-Labrador Time Zone – Newfoundland-Labrador Current Local Time – Daylight Saving Time|publisher=TimeTemperature.com|accessdate=2012-10-26|archive-date=2012-10-23|archive-url=https://web.archive.org/web/20121023120410/http://www.timetemperature.com/tzca/newfoundland_time_zone.shtml|dead-url=no}}</ref> |
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*硫酸两个pKa,分别是−3.0和1.99。<ref>Wenyi Zhao. Handbook for Chemical Process Research and Development. CRC Press, 2016. 2.1.1.2 Sulfuric Acid. ISBN 9781315350202</ref><ref>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</ref> |
*硫酸两个pKa,分别是−3.0和1.99。<ref>Wenyi Zhao. Handbook for Chemical Process Research and Development. CRC Press, 2016. 2.1.1.2 Sulfuric Acid. ISBN 9781315350202</ref><ref>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</ref> |
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*3-氟丙烯的沸点是−3 °C。<ref>Haynes, W. M. (2014). CRC Handbook of Chemistry and Physics 95ed. CRC Press. ISBN 97814822-08689.</ref> |
*3-氟丙烯的沸点是−3 °C。<ref>Haynes, W. M. (2014). CRC Handbook of Chemistry and Physics 95ed. CRC Press. ISBN 97814822-08689.</ref> |
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== 參見 == |
== 參見 == |
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*[[3]] |
*[[3]] |
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* |
*[[-3 dB]](半能點) |
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*[[前 |
*[[前3年]] |
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== 註釋 == |
== 註釋 == |
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第187行: | 第203行: | ||
{{Reflist|2}} |
{{Reflist|2}} |
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== 外部連結 == |
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[[Category:使用创建条目精灵建立的页面]] |
2023年12月17日 (日) 01:22的最新版本
| ||||
---|---|---|---|---|
| ||||
命名 | ||||
小寫 | 負三 | |||
大寫 | 負參 | |||
序數詞 | 第負三 negative third | |||
識別 | ||||
種類 | 整數 | |||
性質 | ||||
質因數分解 | 一般不做質因數分解 | |||
因數 | 1、3 | |||
絕對值 | 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) | |||
在數學中,負三記作−3,是介於負四與負二之間的整數,為3的加法逆元或相反數[1]:22[2],即其與三的和為零[3],偶爾會被視為3的逆反詞或相對概念[4]。日常生活中通常不會用負三來計量事物,例如無法具體地描述何謂負三頭牛[4]或持有負三顆蘋果[5]。
負三經常在訊號處理領域被提及,因為負三分貝約為能量的一半[6]。因此,負三分貝又稱為半能點[7],經常在濾波器、滤光器和放大器[8]中使用[9]。在國際單位制基本單位的表示法中,負三偶爾也會做為冪次來表達立方倒數,比如密度的单位kg・m-3[10]。
性質[编辑]
- 負三為第二大的負奇數。最大的負奇數為負一,而負三為負一的三倍[11]。
- 負三與無理數的值十分接近[12],因此在訊號處理領域中經常使用負三分貝代表能量為一半的情況[6]。
- 負三是最大的負基本判别式[13],同時,在2-rank為0時,負三是絕對值最小的基本判别式[14]。
- 負三能使連續三個奇數的乘積加一為平方數。有這種性質的奇數只有-3和1,而所有滿足n(n+2)(n+4)+1為平方數的整數只有11個,分別為-4, -3, -2, 0, 1, 2, 8, 10, 18, 112, 1272[15]。
- 負三能使二次域的類数為1,即的類数為1,亦即其整數環為唯一分解整環[註 1][16],且這個二次域在複平面上形成了一個六角網格,每個六邊形又可分成6個三角形(三角網格)[17]:289。
- 負三與負三的乘積為正九[27],即負三的平方為九[28],因此負三為九的平方根之一,即九的負平方根。[註 2]
- 現有兩數i和j,i和j的乘積與六倍i和j的和相等,且其和與i、j皆為整數的結果只有8個解,負三是其中之一[31]。
- 負三為四維超立方體(或四維超方形)下闭集合中欧拉示性数的最小值[32]。
負三的因數[编辑]
負三的因數有-3, -1, 1和3[33],這些因數與3的因數相同。在質因數分解中,雖然能夠透過將負一提出來完成質因數分解[34][35], 即,然而算术基本定理一般以探討正整數的質因數分解為主[16],因此一般不會對負的整數進行質因數分解。[36]
負三次冪[编辑]
若一數的冪為負三次,則其可以視為立方的倒數,例如日常生活中常用的密度CGS制單位g/cm3[37],其因此可以表示為質量乘以長度的立方倒數,計為ML-3,此時負三用以表示立方的倒數[38]。
而立方倒數中的相關議題還有立方倒數和。自然數的負三次次方和(立方倒數和)會收斂並趨近於阿培里常数,即:
- = [39]
即全體自然數的負三次方和會收斂在這個數。其值約為1.202056903。同時其也是Zeta函數代入3的結果[39]。
表示方法[编辑]
負三通常以在3前方加入負號表示[1]:28[40],通常稱為「負三」或大寫「負叄」、「負叁」或「負參」,而在某些場合中,會以「零下三」表達-3,例如在表達溫度時[41][42]。而在英語中通常以negative three(負三)表示,比較不會以minus three(減三)表示[43]。
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[44],即將所有位數填上1,再向下減。此時,負三計為「......11111101(2)」,例如,在八位元的二補數二進制中,負三會以「11111101(2)」表示,正三會以「00000011(2)」;而在使用負號的表示法中,負三計為「-11(2)」,亦有在最高位填1表示其為負之表示法,此時負三表示為「10000011(2)」[45]。
在其他領域中[编辑]
- 當分貝數為負三時,能量約為一半,又稱為半能點[7]。
- 智能不足輕度與中度的分界為智力測驗平均值的負三個標準差上[46]
- 關於十的負三次冪10-3 , 其為SI前缀之一,可以用m (Milli)表示。[47]例如:1毫米為10-3 米、1毫克為10-3 克[48]。
- 密度的因次是ML-3,對應的SI制單位可以表示為kg・m-3。[10][51] 而加加速度的因次與單位也能用負三次冪表示,其因次計為LT-3、對應的單位可以用m・s-3 表示 。[52]
- 部分紀年方法或計算機程序[註 3]容許負值的公元年,此時負三年代表的意義為公元前4年[54],同理,對世紀而言負三世紀代表公元前4世纪。[55]
- 《-3℃》為岩井由紀子1987年發行的單曲。[56]
- 火星[57]和木星[58]有時會被歸類在負三等星。此外負三等星亦用於火流星的定義:比負三等星亮的流星稱為火流星[59]。
- 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]
參見[编辑]
註釋[编辑]
參考文獻[编辑]
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