# 記數系統

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 20 24 30 32 36 60 64

• 有效地描述一組數（例如，整數實數
• 所有的數對應唯一的表示（至少有一個標準表示法）
• 反映數的代數算術結構

## 進位制詳解

${\displaystyle (a_{n}a_{n-1}...a_{1}a_{0}.c_{1}c_{2}c_{3}...)_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}}$

## 進位轉換

1020304 / 7 = 145757 r 5 ↑  => 11446435
145757 / 7 =  20822 r 3 │
20822 / 7 =   2974 r 4 │
2974 / 7 =    424 r 6 │
424 / 7 =     60 r 4 │
60 / 7 =      8 r 4 │
8 / 7 =      1 r 1 │
1 / 7 =      0 r 1 │

10110111 / 101 = 100100 r 11  (3) ↑  => 1213
100100 / 101 =    111 r  1  (1) │
111 / 101 =      1 r 10  (2) │
1 / 101 =      0 r  1  (1) │

0.1A4C × 9 = 0.ECAC │
0.ECAC × 9 = 8.520C │
0.520C × 9 = 2.E26C │
0.E26C × 9 = 7.F5CC │
0.F5CC × 9 = 8.A42C │
0.A42C × 9 = 5.C58C ↓  => 0.082785...a

## 註釋

1. ^ David Eugene Smith; Louis Charles Karpinski. The Hindu-Arabic numerals. Ginn and Company. 1911.

## 參考文獻

• Georges Ifrah. The Universal History of Numbers : From Prehistory to the Invention of the Computer, Wiley, 1999. ISBN 0-471-37568-3
• 高德納. 《電腦程式設計藝術》. Volume 2, 3rd Ed. Addison-Wesley. pp.194–213, "Positional Number Systems".
• J.P. Mallory and D.Q. Adams, Encyclopedia of Indo-European Culture, Fitzroy Dearborn Publishers, London and Chicago, 1997.
• A.L. Kroeber (Alfred Louis Kroeber) (1876–1960), Handbook of the Indians of California, Bulletin 78 of the Bureau of American Ethnology of the Smithsonian Institution (1919)
• Hans J. Nissen; Peter Damerow; Robert K. Englund. Archaic Bookkeeping: Early Writing and Techniques of Economic Administration in the Ancient Near East. University Of Chicago Press. 1993. ISBN 978-0-226-58659-5.
• Schmandt-Besserat, Denise. How Writing Came About. University of Texas Press. 1996. ISBN 978-0-292-77704-0.
• Zaslavsky, Claudia. Africa counts: number and pattern in African cultures. Chicago Review Press. 1999. ISBN 978-1-55652-350-2.nd}}