Base 24

Base 24

The base-num|24 system is a numeral system with 24 as its base.

There are 24 hours in a day, so our time keeping system includes a base-24 component. See also base 12. Decimal Equivalent 10 twenty four 24 24 100 ? 24^2 = 576 1000 ? 24^3 = 13 824 10 000 ? 24^4 = 331 776 100 000 ? 24^5 = 7 972 624 1 000 000 ? 24^6 = 191 102 976

The digits used for numerals ten (10) to twenty three (23) may be the letters "A" through to "P" ("I" and "O" are skipped to prevent confusion with the digits 1 and 0 in some typefaces).

Fractions

Quadrovigesimal fractions are usually either very simple

1/2 = 0.C 1/3 = 0.8 1/4 = 0.6 1/6 = 0.4 1/8 = 0.3 1/9 = 0.2G 1/C = 0.2 1/G = 0.1C 1/J = 0.18

or complicated

1/5 = 0.4K4K4K4K... recurring (easily rounded to 0.5 or 0.4K) 1/7 = 0.3A6LDH3A6... recurring 1/A = 0.29E9E9E9... recurring (rounded to 0.2A) 1/B = 0.248HAMKF6D248.. recurring (rounded to 0.24) 1/D = 0.1L795CN3GEJB1L7.. recurring (rounded to 0.1L) 1/P = 0.11111... recurring (rounded to 0.11) 1/11 = 0.0P0P0P... recurring (rounded to 0.0P) (1/(5*5))

As explained in recurring decimals, whenever a fraction is written in "decimal" notation, in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base. Thus, in base-10 (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: ¹⁄8 = ¹⁄(2*2*2), ¹⁄20 = ¹⁄(2×2×5), and ¹⁄500 (2²×5³) can be expressed exactly as 0.125, 0.05, and 0.002 respectively. ¹⁄3 and ¹⁄7, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, ¹⁄8 is exact; ¹⁄20 and ¹⁄500 recur because they include 5 as a factor; ¹⁄3 is exact; and ¹⁄7 recurs, just as it does in base 10.

In practical applications, the nuisance of recurring decimals is encountered less often when quadrovigesimal (or duodecimal) notation is used.

However when recurring fractions do occur in quadrovigesimal notation, they sometimes have a very short period when they are numbers containing one or two factors of five, as 5² = 25 is adjacent to 24. The other adjacent number, 23, is a prime number. So certain powers of five are palindromes in the quadrovigesimal notation:

51 = 5 52 = 11 53 = 55 54 = 121 55 = 5A5 56 = 1331 57 = 5FF5 58 = 14641

The multiples of decimal hundred are 44, 88, CC, GG, LL, 110, etc.

Natural languages

Umbu-Ungu, [ citation
contribution=Umbu-Ungu
title=Ethnologue: Languages of the World
edition=15
url=http://www.ethnologue.com/show_language.asp?code=ubu
editor-last=Gordon
editor-first=Raymond G., Jr.
year=2005
accessdate=2008-03-16] also known as Kakoli, is reported to have base-24 numerals [ Citation
last=Bowers
first=Nancy
last2=Lepi
first2=Pundia
title=Kaugel Valley systems of reckoning
year=1975
journal=Journal of the Polynesian Society
volume=84
issue=3
pages=309-324
url=http://www.ethnomath.org/resources/bowers-lepi1975.pdf] . "Tokapu" means 24, "tokapu talu" means 24×2 = 48, and "tokapu tokapu" means 24×24 = 576.

References


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