五進制
五進制(英文:quinary、base-5、pental[1][2][3])係以5做底嘅進位制,只係用到0、1、2、3、4五個數字記數。呢種進位制有可能源自人一隻手有5隻手指,而家只有少數語言用呢種進位制。
喺呢個進位制入邊,平時嘅5會寫做10,平時嘅25會寫做100,平時嘅60寫做220,等等。
由於5係質數,分數嚟講只有啲分母係5嘅次方嘅會係有限小數,不過因為5喺兩個高合成數(4同6)中間,好多分數嘅循環節都好短。
今時今日,大部份用5做底嘅記數系統都係所謂嘅二五進制(biquinary),實則係十進制,不過用5作為一個sub-base。另一個sub-base記數系統係大家熟悉嘅六十進制,用10做sub-base。平時用嚟記時嘅時分秒系統就係用呢一隻記數法。
五進制入邊每一個位可以表達位元嘅資訊。
同其他進制嘅比較
編輯× | 1 | 2 | 3 | 4 | 10 | 11 | 12 | 13 | 14 | 20 |
1 | 1 | 2 | 3 | 4 | 10 | 11 | 12 | 13 | 14 | 20 |
2 | 2 | 4 | 11 | 13 | 20 | 22 | 24 | 31 | 33 | 40 |
3 | 3 | 11 | 14 | 22 | 30 | 33 | 41 | 44 | 102 | 110 |
4 | 4 | 13 | 22 | 31 | 40 | 44 | 103 | 112 | 121 | 130 |
10 | 10 | 20 | 30 | 40 | 100 | 110 | 120 | 130 | 140 | 200 |
11 | 11 | 22 | 33 | 44 | 110 | 121 | 132 | 143 | 204 | 220 |
12 | 12 | 24 | 41 | 103 | 120 | 132 | 144 | 211 | 223 | 240 |
13 | 13 | 31 | 44 | 112 | 130 | 143 | 211 | 224 | 242 | 310 |
14 | 14 | 33 | 102 | 121 | 140 | 204 | 223 | 242 | 311 | 330 |
20 | 20 | 40 | 110 | 130 | 200 | 220 | 240 | 310 | 330 | 400 |
五進制 | 0 | 1 | 2 | 3 | 4 | 10 | 11 | 12 | 13 | 14 | 20 | 21 | 22 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
二進制 | 0 | 1 | 10 | 11 | 100 | 101 | 110 | 111 | 1000 | 1001 | 1010 | 1011 | 1100 |
十進制 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
五進制 | 23 | 24 | 30 | 31 | 32 | 33 | 34 | 40 | 41 | 42 | 43 | 44 | 100 |
二進制 | 1101 | 1110 | 1111 | 10000 | 10001 | 10010 | 10011 | 10100 | 10101 | 10110 | 10111 | 11000 | 11001 |
十進制 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
十進制 (循環節) | 五進制 (循環節) | 二進制 (循環節) |
1/2 = 0.5 | 1/2 = 0.2 | 1/10 = 0.1 |
1/3 = 0.3 | 1/3 = 0.13 | 1/11 = 0.01 |
1/4 = 0.25 | 1/4 = 0.1 | 1/100 = 0.01 |
1/5 = 0.2 | 1/10 = 0.1 | 1/101 = 0.0011 |
1/6 = 0.16 | 1/11 = 0.04 | 1/110 = 0.010 |
1/7 = 0.142857 | 1/12 = 0.032412 | 1/111 = 0.001 |
1/8 = 0.125 | 1/13 = 0.03 | 1/1000 = 0.001 |
1/9 = 0.1 | 1/14 = 0.023421 | 1/1001 = 0.000111 |
1/10 = 0.1 | 1/20 = 0.02 | 1/1010 = 0.00011 |
1/11 = 0.09 | 1/21 = 0.02114 | 1/1011 = 0.0001011101 |
1/12 = 0.083 | 1/22 = 0.02 | 1/1100 = 0.0001 |
1/13 = 0.076923 | 1/23 = 0.0143 | 1/1101 = 0.000100111011 |
1/14 = 0.0714285 | 1/24 = 0.013431 | 1/1110 = 0.0001 |
1/15 = 0.06 | 1/30 = 0.013 | 1/1111 = 0.0001 |
1/16 = 0.0625 | 1/31 = 0.0124 | 1/10000 = 0.0001 |
1/17 = 0.0588235294117647 | 1/32 = 0.0121340243231042 | 1/10001 = 0.00001111 |
1/18 = 0.05 | 1/33 = 0.011433 | 1/10010 = 0.0000111 |
1/19 = 0.052631578947368421 | 1/34 = 0.011242141 | 1/10011 = 0.000011010111100101 |
1/20 = 0.05 | 1/40 = 0.01 | 1/10100 = 0.000011 |
1/21 = 0.047619 | 1/41 = 0.010434 | 1/10101 = 0.000011 |
1/22 = 0.045 | 1/42 = 0.01032 | 1/10110 = 0.00001011101 |
1/23 = 0.0434782608695652173913 | 1/43 = 0.0102041332143424031123 | 1/10111 = 0.00001011001 |
1/24 = 0.0416 | 1/44 = 0.01 | 1/11000 = 0.00001 |
1/25 = 0.04 | 1/100 = 0.01 | 1/11001 = 0.00001010001111010111 |
語言
編輯有唔少嘅語言[4]都仲用緊五進制,例如Gumatj、Nunggubuyu[5]、Kuurn Kopan Noot[6]、Luiseño[7]同Saraveca等。Gumatj係真正嘅「5-25」語言,入邊 25 係比 5 大嘅「單位」,下邊係 Gumatj 嘅數字[5]:
十進制數字 | 五進制 | Gumatj 入邊嘅數字 |
---|---|---|
1 | 1 | wanggany |
2 | 2 | marrma |
3 | 3 | lurrkun |
4 | 4 | dambumiriw |
5 | 10 | wanggany rulu |
6 | 11 | wanggany rulu ga wanggany |
7 | 12 | wanggany rulu ga marrma |
8 | 13 | wanggany rulu ga lurrkun |
9 | 14 | wanggany rulu ga dambumiriw |
10 | 20 | marrma rulu |
15 | 30 | lurrkun rulu |
20 | 40 | dambumiriw rulu |
25 | 100 | dambumirri rulu |
50 | 200 | marrma dambumirri rulu |
75 | 300 | lurrkun dambumirri rulu |
100 | 400 | dambumiriw dambumirri rulu |
125 | 1000 | dambumirri dambumirri rulu |
625 | 10000 | dambumirri dambumirri dambumirri rulu |
雙五進制
編輯用2、5做sub-base嘅十進制系統又叫做雙五進制(biquinary),Wolof同Khmer都用呢隻系統。羅馬數字都算係用呢種系統,例如7寫做VII,即係5+1+1,70就係LXX,即係50+10+10。成個數字表係:
I | V | X | L | C | D | M |
1 | 5 | 10 | 50 | 100 | 500 | 1000 |
值得留意嘅係,呢個系統唔係進位制嚟。理論上一個數字可以用任何次序表達,例如73可以寫做LXXIII同IIIXXL都得,不過其實羅馬數字仲有一條「調轉次序代表減數」嘅規則,即係IV係4,IX係9咁樣,所以羅馬數字有一個既定嘅次序去寫出嚟。
好多算盤都用雙五進制嚟方便運算,例如中國算盤、日本算盤咁樣。記數符號(tally mark)、劃正字等等都可以算係用緊五進制/雙五進制。好多貨幣都係行緊雙五進制或者係部份雙五進制,例如港紙有5亳、5蚊、50蚊、500蚊呢啲幣值。
四五進制
編輯計數機同程式語言
編輯有少數嘅計數機可以用五進制嚟計數,例如聲寶嗰啲,包括EL-500W同EL-500X系列,入邊佢叫做pental system[1][2][3]。另外開源計數機WP 34S都有用五進制。
Python嘅int()
函數可以將任何進制嘅數轉返做十進制,例如五進制嘅101可以用int('101',5)
轉返做十進制嘅26[9]。
睇埋
編輯參考資料
編輯- ↑ 1.0 1.1 "Archived copy" (PDF). 原先內容歸檔 (PDF)喺2017-07-12. 喺2017-06-05搵到.
{{cite web}}
: CS1 maint: archived copy as title (link) - ↑ 2.0 2.1 "Archived copy" (PDF). 原先內容歸檔 (PDF)喺2016-02-22. 喺2017-06-05搵到.
{{cite web}}
: CS1 maint: archived copy as title (link) - ↑ 3.0 3.1 "Archived copy" (PDF). 原先內容歸檔 (PDF)喺2017-07-12. 喺2017-06-05搵到.
{{cite web}}
: CS1 maint: archived copy as title (link) - ↑ Harald Hammarström, Rarities in Numeral Systems: "Bases 5, 10, and 20 are omnipresent."
- ↑ 5.0 5.1 Harris, John (1982), Hargrave, Susanne (編), "Facts and fallacies of aboriginal number systems" (PDF), Work Papers of SIL-AAB Series B, 8: 153–181, 原著 (PDF)喺2007-08-31歸檔
- ↑ Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
- ↑ Closs, Michael P. Native American Mathematics. ISBN 0-292-75531-7.
- ↑ "Numbers in Nahuatl". omniglot.com. 喺2022-05-19搵到.
- ↑ "Convert base-2 binary number string to int". Stack Overflow. 原先內容歸檔喺24 November 2017. 喺5 May 2018搵到.