查看“︁底數經濟度”︁的源代码

'''底數經濟度'''是一種將某數作為[[進位制]]的[[底数 (进制)|底数]]時，該進位制表達數的效率，其定義為數在某進位制下表達的位數與底數或該計數系統中每個位數可能的符號之數量的乘積。為量化不同底數的進制或計數系統在表示一個數時的效率的一種方法，尤其用於計算機系統，評估特定計數系統的儲存效率。

底數經濟度的概念亦用於[[組織結構]]、網路等領域。

== 定義 ==
對一數N在特定的底數b下，'''底數經濟度''' <math>E(b,N)</math>定義為：
: <math>E(b,N) = b \lfloor \log_b (N) +1 \rfloor \,  </math>
其中，<math>\lfloor \, \rfloor</math>表示[[取整函数|下取整函數]]；<math>\log_{b}</math>表示以<math>b</math>為底的對數。

若b和N皆為正整數，則底數經濟度<math>E(b,N)</math>值與<math>N</math>在以<math>b</math>為底的進制下的位數與<math>b</math>的乘積<ref name="Hayes">{{cite journal | title = Third Base | author = [[Brian Hayes (scientist)|Brian Hayes]] | journal = [[American Scientist]] | volume = 89 | issue = 6 | year = 2001 | pages = 490 | doi = 10.1511/2001.40.3268 | url=http://www.americanscientist.org/issues/pub/2001/11/third-base | accessdate=2013-07-28 | archive-url = https://web.archive.org/web/20140111055213/http://www.americanscientist.org/issues/pub/2001/11/third-base |  | archive-date = 2014-01-11 | dead-url = yes }}</ref>。

== 底數經濟度列表 ==
:{| class="wikitable sortable"
|-
! [[底数_(进制)|底数]] ''b''
! ''N'' = 1 to 6
''E''(''b'',''N'')平均
! ''N'' = 1 to 43
''E''(''b'',''N'')平均
! ''N'' = 1 to 182
''E''(''b'',''N'')平均
!''N'' = 1 to 5329
''E''(''b'',''N'')平均
! <math> {{E(b)} \over {E(e)}} </math>
! ''E''&thinsp;(''b''&thinsp;)''/E''&thinsp;(''e''&thinsp;)的<br/>相對大小
|- align="right"
| [[一進制|1]]
| 3.5
| 22.0
| 91.5
| 2,665.0
| <math> \infty </math> || align="left"|&mdash;
|- align=right
| [[二進制|2]]
| 4.7
| 9.3
| 13.3
| 22.9
| {{bartable|1.0615||20}}
|- align=right
| ''[[E进制|e]]''
| 4.5
| 9.0
| 12.9
| 22.1
| {{bartable|1.0000||20}}
|- align=right
| [[三進制|3]]
| 5.0
| 9.5
| 13.1
| 22.2
| {{bartable|1.0046||20}}
|- align=right
| [[四進制|4]]
| 6.0
| 10.3
| 14.2
| 23.9
| {{bartable|1.0615||20}}
|- align=right
| [[五進制|5]]
| 6.7
| 11.7
| 15.8
| 26.3
| {{bartable|1.1429||20}}
|- align=right
| [[六進制|6]]
| 7.0
| 12.4
| 16.7
| 28.3
| {{bartable|1.2319||20}}
|- align=right
| {{link-en|七進制|Septenary|7}}
| 7.0
| 13.0
| 18.9
| 31.3
| {{bartable|1.3234||20}}
|- align=right
| [[八進制|8]]
| 8.0
| 14.7
| 20.9
| 33.0
| {{bartable|1.4153||20}}
|- align=right
| [[九進制|9]]
| 9.0
| 16.3
| 22.6
| 34.6
| {{bartable|1.5069||20}}
|- align=right
| [[十進制|10]]
| 10.0
| 17.9
| 24.1
| 37.9
| {{bartable|1.5977||20}}
|- align=right
| [[十二進制|12]]
| 12.0
| 20.9
| 25.8
| 43.8
| {{bartable|1.7765||20}}
|- align=right
| {{link-en|十五進制|Pentadecimal|15}}
| 15.0
| 25.1
| 28.8
| 49.8
| {{bartable|2.0377||20}}
|- align=right
| [[十六進制|16]]
| 16.0
| 26.4
| 30.7
| 50.9
| {{bartable|2.1230||20}}
|- align=right
| [[二十進制|20]]
| 20.0
| 31.2
| 37.9
| 58.4
| {{bartable|2.4560||20}}
|- align=right
| 30
| 30.0
| 39.8
| 55.2
| 84.8
| {{bartable|3.2449||20}}
|- align=right
| 40
| 40.0
| 43.7
| 71.4
| 107.7
| {{bartable|3.9891||20}}
|- align=right
| [[六十進制|60]]
| 60.0
| 60.0
| 100.5
| 138.8
| {{bartable|5.3910||20}}
|}

==參考文獻==
{{Reflist}}

==延伸閱讀==
*S.L. Hurst, "Multiple-Valued Logic-Its Status and its Future", ''IEEE trans. computers'', Vol. C-33, No 12, pp.&nbsp;1160&ndash;1179, DEC 1984.
*J. T. Butler, "Multiple-Valued Logic in VLSI Design, ” IEEE Computer Society Press Technology Series, 1991.
*C.M. Allen, D.D. Givone “The Allen-Givone Implementation Oriented Algebra", in ''Computer Science and Multiple-Valued Logic: Theory and Applications'', D.C. Rine, second edition, D.C. Rine, ed., The Elsevier North-Holland, New York, N.Y., 1984. pp.&nbsp;268&ndash;288.
*G. Abraham, "Multiple-Valued Negative Resistance Integrated Circuits", in ''Computer Science and Multiple-Valued Logic: Theory and Applications'', D.C. Rine, second edition, D.C. Rine, ed., The Elsevier North-Holland, New York, N.Y., 1984. pp.&nbsp;394&ndash;446.

==外部連結==
*[https://web.archive.org/web/20170421123129/http://gtode.tripod.com/Optimum.htm The Optimum Radix In Multiple-Valued Logic Systems]

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